 When we count, we make an abstraction of a property of a set. That property might be called the number of elements. We'll discuss that in more detail later, but for now let's consider counting, and an important idea to keep in mind in all the discussion that follows. Always distinguish between easy and familiar. Now in order to be able to describe quantity, we'll need words for each size. But there's a problem, and remember always distinguish between easy and familiar. Suppose you counted this way. 1, 2, 3, 4, 5, 6, and so on. No word is related to any other, and you don't know what comes next. And so here's the important question. How many different words do you want for numbers? Well, let's go to an extreme. What if you only had a word for one item? Now remember, a problem exists independent of your ability to solve it. The problem of describing a large quantity of objects is going to exist whether or not we have the words for the large quantity. We still need to find a solution, and so we could describe larger amounts using a compound word, a word formed by stringing already existing words together. So this amount is a, well, how about a 1, 1? And if we have more, this is a 1, 1, 1, and a 1, 1, 1, and so on. And so we see that it would be possible to express the amount shown by repeatedly using the word 1. But it's easier if you have words for larger amounts. So suppose we have words for the amounts 1, 2, 3, and 4, but no words for higher numbers. Again, you might know these words for higher numbers, but remember, always distinguish between easy and familiar. These words are familiar, but they're not easy. It is literally like learning a foreign language, because at some point you didn't know what these words were and you had to learn them. We could express larger amounts using just these words. So here we have a 4, here's a 4 and 1, a 4 and 2, a 4 and 3, a 4 and 4, and 4, and 3, and so on. Now an important idea is to remember how you speak influences how you think. Suppose you have words for 1, 2, 3, and 4, but no higher amounts. In effect, if we have more than 4, we have lots. But we can count the lots by counting out sets of 4, and this leads to the idea of a unit. A unit is a single object or a set of objects we treat as a single object. For example, a hot dog is a unit, a pack of hot dogs is a unit, and a case of hot dogs is a unit. Sometimes our units are irregular. For example, 8 ounces is a unit, which we call a cup. But 4 cups is another unit, which we call a quart. But it's actually easier if our units are regular, so 12 of anything is a unit, a dozen. 12 dozen is a unit, a gross. 12 gross is a unit, a great gross, and so on. We say a system is base n if any set of n makes a unit. And just like the words for numbers, we have to come up with the names for the units. And just like the words for numbers, the names of the units are also random sounds. For example, we might call our units 1, Dec, Hecht, Kilo, Miri, or maybe small, medium, large, extra large, huge, ginormous, or anything else we agree upon. So let's try to describe the first three units in base 6. So in base 6, 6 of anything make up the next larger unit. So a single object is a unit. Six objects make a unit. Six of these make a larger unit. And we might have names for these, so we might call these smallest set a 1. The next larger set we might call a 6. The next larger set we'll call a... well, anyway, we don't know what they are, but the larger units exist even if we don't have names for them. While we could write everything out in words, it's convenient to have special symbols. For the first few numbers, we might have the symbols... Well, again, the symbols are arbitrary, so we don't want to have too many. Suppose you're working base 6, what symbols do you need? And so the important idea here is that we want a way to express any amount we encounter. So we would need symbols for the amounts 1, 2, 3, 4, and 5. And we'll use our standard symbols for these. What about 6? Since we're working base 6, we can trade 6 of anything for 1 of the next larger unit. And what this means is we don't need a symbol for 6. Our common system of numeration is base 10, so any set of 10 makes 1 of the next larger unit. We also need symbols for amounts up to 10, and so we use these. Now, we don't need symbols for the units, but it's convenient to at least have names for them. And so we often use the following names. The smallest unit is called a 1. 10, because we're working base 10, makes another unit called a 10. 10 tens make 100. 10 hundreds make 1,000. And we can continue, but the next few units don't have their own names. So 10,000 is a unit, but we just call it a 10,000. Where 10,000 is not really a unit name, but rather it's a compound name 10,000. And likewise, 10 tens make a unit, but we just call it a 100,000. And 10 hundreds make a new unit. And in this case, we finally do introduce a new name for a unit. We call this a million.