 Alright, so let's take a look at one more extension of our number system, and that's to the decimal realm. So in base N, each unit is N of the next unit. So for example, if I'm working in base 10, I have my unit, which I'm going to call a 1, there's our smallest amount. Ten of these make a 10, and that's our next size unit. Ten of those make a hundred, and that's our next size unit. And so on, as far as we want to name larger and larger units. On the other hand, I can imagine finding smaller units. So for example, let's blow that one up. So it's a big thing like this, and there's my one again. I've just drawn it larger so that I can divide it into smaller pieces, and I can break it into ten pieces. Well, I would call each of these pieces a tenth, and I might take one of those tenths, and break those into even smaller pieces, and I'm going to call each of these a... Well, we need a name for them, so to remember that ten of the original pieces formed a tenth, and so ten tenths form a one. So each of these very small pieces here, well if I take each of these small pieces here, turns out that one hundred of them are going to form a one, so I'll call them a hundredth. And I can keep going. I can break each of those hundredths into ten more pieces, and each piece would be called a thousandth, and again, and as far as I care to invent new names for smaller and smaller pieces. Now, let's talk about naming these numbers. Consistency counts. There's really no difference whatsoever between expressing a number using tens, hundreds, and so on. We're just extending our place value chart. So for example, let's take a look at a number here, and I have my place value chart where I'll actually write out what the names of the units are, and I have some number of hundreds, some number of tens, some number of tens, some number of hundreds, and so on. So I can express the number in words by identifying how many of each unit. So here I have one hundred, two tens, one ten, three hundreds, and so I should call the number one hundred, twenty, one ten, three hundreds. And that would be absolutely consistent with how we name numbers, except English wins. Unfortunately, English, as well as all other natural languages, is very inconsistent when we name amounts smaller than one. And how we do that is the following. We identify the whole number and the fractional part. We split them using the word and. The fractional part, for reasons unknown, but again, inconsistency of natural languages, is going to be traded into our smallest unit. So what does that mean in this case? So my whole number part is still going to be one hundred, two tens, one hundred, twenty. The fractional part I'm going to express by saying and, whatever this is, and the other thing I'm going to do is I'm going to pile all these pieces into the smallest available unit. Well, I can do that using the trade rate. I'm still working base ten. So my trade rate is still ten for one. So this tenth here, this thing here trades for ten more in the next place over. And so what I have is this as the amount that I'm going to try and name, and I'll name it one hundred, two tens, and how many things do I have here? I have a total of thirteen hundreds. So I'm going to name this thirteen hundreds. And so there's my name for the number one hundred twenty and thirteen hundreds. So far, so good. But this is the verbal expression of the number. We'd actually like to try and write what our number is. So again, we want to take our number here. This is what we're actually working with. We don't have an abstract symbol for thirteen. What we have to do is we only have abstract symbols for the amounts from zero up through nine. We're going to put that tenth back where it belongs. And we're going to write our number by replacing the concrete objects with our abstract number symbols. So what's that going to look like? Well, there's one. This concrete object here corresponds to the abstract number symbol two. There's nothing here, so my abstract symbol is going to be zero. There's this, which I'll replace with the corresponding abstract symbol one. And there's this, which I'll replace with the corresponding abstract symbol three. And then finally, I'll drop the place value chart to get our numerical value. Except, as written, I can't distinguish this number from twelve thousand no hundreds ten and three. So I have no way of distinguishing this number, which is actually one hundred two tens and thirteen hundreds from a much larger number. To avoid this confusion, what I'm going to do is I'm going to use a decimal point to separate the ones from the tenths. And so this number becomes one hundred twenty. And the decimal point tells us that here's the ones place. This is one hundred twenty. There's our whole number portion and thirteen hundreds.