 An important idea in elementary mathematics is the idea of a unit. A unit is a single object or a set of objects we treat as a single object. For example, an egg is a unit, but twelve eggs is also a unit which we happen to call a dozen. Similarly, a year is a unit of time, but one hundred years is also a unit of time which we call a century. A cent is a unit of money, and twenty-five cents is another unit of money which we call a quarter. So it's helpful to remember that arithmetic is bookkeeping. We're trying to keep track of how many of which units. Now in our system of writing numbers, our units are ones. A unit consisting of ten ones, which we call a ten. A unit consisting of ten tens, which we call a hundred. A unit consisting of ten hundreds, a thousand, and so on, even though we might not have names for the higher units. Since each unit is ten times the size of the previous unit, we say that our system is base ten. So when we write a number, we record how many of which units, with the smallest units corresponding to the right most place. So we should record the units. So maybe we have one hundred, two tens, and three ones. But as long as we're agreed on what the base is, we can omit the units, and write this as one, two, three. Numbering in a different base might seem exotic, but there are bases we commonly use in specific situations. So for example, consider the coins penny, nickel, quarter. Each of these is five of the smaller. So five pennies are equivalent to one nickel. Five nickels are equivalent to one quarter. Since each is five of the smaller, this is base five. Another common situation we encounter is when we are dealing with things like eggs. We have a single unit, a dozen, and a gross. Each of these is twelve of the smaller, and so this is an example of base twelve. As one of the more important bases, consider the time units, second, minute, hour. Each is sixty of the smaller, and so this is base sixty. An important idea to keep in mind is that the base is independent of the item counted. We see base five in currency, but we can use base five to count the days of the week. And while we ordinarily see base twelve when counting eggs, we can use base twelve to count money. And even though base sixty is most familiar when we talk about time, we can use base sixty to count people. So what if we're using a different base? As long as we write the units, it doesn't matter. So for example, if I write down one quarter, two nickels, three pennies, this amount is obviously different from one hundred to tens, three ones. And if we drop the units, how can we distinguish between them? And so to avoid confusion, we usually spell the base out. So one quarter, two nickels, three pennies, well that's really one, two, three, base five. Meanwhile one hundred, two tens, three ones, well that's one, two, three, base ten. Now since base ten is our usual number system, we might omit the ten. So we would write one, two, three, base ten, just as one, two, three. Because base sixty shows up often enough, there are three common ways to write a number in base sixty, and it's useful to remember numbers in base sixty are like recording time in seconds, minutes, and hours. So suppose we have five hours, ten minutes, thirty seconds. We could write this in a number of different ways. We could separate the places with commas and write five, comma, ten, comma, thirty. The comma separator is the most common in mathematics, but in everyday life we often see slightly different forms. So another common way of representing numbers in base sixty is to separate the places with colons, five, colon, ten, colon, thirty. And this is what you see on a digital clock. And on train and airport schedules you might also see all of the numbers run together using two digits for any single digit amount, so o five, one zero, three zero, with nothing separating the places. And again remember the base is independent of the item counted. A number in base sixty could be a length, an area, a number of people, or an amount of money. Now it's sometimes helpful to get an idea of what these numbers are, so let's interpret four, comma, thirty. Because of the way it's written we should assume this is base sixty, and so four, comma, thirty. Well the right most place corresponds to the ones, so that's thirty ones. The next place over is a sixty, so that's four sixties. So four, comma, thirty would be four sixties, thirty ones. And what's important here is this is how you really should read it. This is four sixties, thirty ones, and that's all there is to the interpretation. But you might want to know what that is in a more familiar form, so let's take a look at that. Four sixties is what we ordinarily think about as two hundred forty. And thirty ones is what we ordinarily think about as, well, thirty. Arithmetic is bookkeeping. So when we've recorded that we have four sixties and thirty ones, that's another way of saying that we have two forty plus thirty, two hundred seventy. We can also go the other way and express a number in base sixty, like one hundred and fifty. So remember arithmetic is bookkeeping how many of which units? In base sixty our units are going to be ones and sixties. So let's take a look at that one hundred and fifty. Since one hundred and fifty is more than sixty, we know there's going to be a couple of sixties, and in fact if I take two sixties, that's one hundred and twenty, and that means I have thirty ones left over. So that says two sixties and thirty ones is the same as one hundred and fifty. And again arithmetic is bookkeeping. We want to record how many of which units? So in base sixty, one hundred and fifty would be two sixties and thirty ones, which we write as two comma thirty. Now all of the preceding is useful to convert back and forth between base ten and base sixty, but actually there's no good reason to switch between bases. And a useful idea to remember is that if you're given numbers in base sixty, you should compute with numbers in base sixty.