 Well, at some point we're going to want to switch from words to mathematical symbols, and so this comes down to the question of, well, how do I write the numerical amount? So again, let's consider our problem where we have a bunch of things in here, and what I'd like to do is I'd like to express the number of elements of this set in base 8. So previously, we determined what we had in this set in base 8. I can describe that by finding my sets of 8. There's 2, 8s, and 4, and what I want to do now is I want to translate the words 2, 8s, and 4 into a numerical value that I can express. So I'm going to go through that in two steps. So to begin with, we have the words 2, 8s, and 4, and again, in deference to our colleagues in English, we will omit the word and, even though almost everybody wants to put that in there, and so the cardinality there, 2, 8s, 4. Now life is easiest when we are consistent, and what we are inconsistent about here is that 8 is a unit in base 8, and so this is saying we have two of these 8. Here's one of them, here's another one. We have 4, but there's no unit word attached to this, and so we might want to be consistent and attach the unit word here. This is 4 of the single units, so we might say that what we have is 2, 8s, 4, 1s, and so we have how many of each type of unit? We have 2, 8s, we have 4, 1s, and now my expression of the written word is consistent. And now we're ready to move on to expressing the values. So in base 8, we'll have the symbols for the amounts from 0 up to 7. We'll always have symbols for the amounts from 0 up to 1 less than the base. So if I'm working base 8, I'll have symbols from 0 through 7, and what this means is I can replace the number words for 0 through 7 with the corresponding number symbols for 0 through 7. And to avoid over-complicating things, we'll just use our standard symbols. I know how to write 2, and I know how to write 4. And I have my partial transition to a purely symbolic expression of the numerical amount. Now some authors call this a dispositional form, and the reason for that is that we are still specifying what our units are. We have 2, 8s, we have 4, 1s, and anybody seeing this knows exactly what we're talking about. They may think that it's a little bit strange, and know how we're expressing it, but if you go up to somebody on the street and say, I have 2, 8s, and 4, 1s, they can, once they figure you're talking about a numerical quantity, they'll understand that you're looking at something like this. And so there's a dispositional form. Our final transition to a purely symbolic form is to drop the unit names. So they're gone. And to remind ourselves that we are actually working in base 8, we like to spell out the base down below as a subscript. So our cardinality here would be 2, 4, base 8. And it's very important to, when you read this, to read this as 2, 4, base 8. There is a temptation to read this as a quote, unquote, normal number. However, this is not a normal number. This is a number that has been expressed in base 8. And it is different from things we've seen before.