 So fractions are sometimes regarded as this mysterious new entity that you have to deal with. But the reality is, if you understand what you're doing with the whole numbers, then fractions are identical. There is no difference between the arithmetic of fractions and the arithmetic of whole numbers. How do we mean by that? Well, let's take a look at that. When I write whole numbers, what we do is record the number of each type of unit. So if I'm using, for example, this little for my smallest unit, this for the next larger, this for the largest, and so on, then if I write down a number like 1, 2, 3, who cares what the base is, then what I'm saying is reading from right to left. I have three of the smallest unit, two of the next larger unit, and one of the largest unit there. So this is an abstract representation of the quantity this. Three of the smallest, two of the next larger, one of the next larger. Now that goes a little bit farther. When I'm working in any system, I can always take one of these units and I can break it into some number of smaller pieces. And the number of smaller pieces is going to depend on what the base is. Well, here's a logical question to ask. Well, if I can break this into smaller pieces and I can break this into smaller pieces, why can't I break these small things into smaller pieces as well? And if you are really a mathematician, you say, why not? Let's see what happens. Now, we could break this into a whole bunch of little tiny pieces, but then you'd be straightening at the screen trying to see what those pieces look like. So we'll expand it. So this little piece here, we're going to blow it up into something that's a little bit larger so that when we break it into smaller pieces, we'll be able to see what those pieces are. Now, a couple of terms have to be introduced. If I break this into n smaller pieces, each of the same size, each one of those pieces is going to be called an nth. One nth, whatever n happens to be. And I'm going to express a collection. Well, I express this collection by saying how many of each type of piece we have. I have one of these, two of these, three of these. So I write it as one, two, three, whatever the base is. And if I have a collection of these smaller pieces, I'm going to tell you how many of those smaller pieces I have. For example, let's take a large square and we'll assume that this large square represents a single unit. Now, how do I express the amount that is shaded? So let's go ahead and take this apart. First of all, we have to count. The square has been broken into one, two, three equal pieces. And so that means each piece is one-third. Well, that's not grammatical. We'll say that each part is one-third. So each of these is one-third. And the amount that I have, I have to count, I have two, one, two, that are shaded. And so the amount that I have, I tell you how many and what the units are. There's two of these and each one of them is a third. So I have one, I have two-thirds. So the amount we have is two-thirds. Now, when I go to write that amount, when we write numbers, we drop the unit names or symbols and record only the number. So here I have this thing. I might, as my intermediate step, note that I have one of these, two of these, three of these, one, two, three. And then I'll drop those symbols out. I have one, two, three base, whatever the base happens to be. So in my final written form, I don't have any of my concrete symbols there. Well, I'm going to do exactly the same thing with fractions. So in this amount, what I have is two of these things. And while I do want to name what these things are, so I could write this as two-thirds. And the spelled out third here is a reference to the fact that each one of these rectangles is one-third. But instead, I'm going to drop the unit designation and write this as two over three. And so I have this form of my fraction. Now, there's some important terms here. The denominator names what the unit is. So here my units are thirds. My units are thirds. There's three of them. And so that's going to be the value in the denominator. This number here tells you how big each piece is. The other component we have is the numerator tells us how many we have. It numbers them. I have two one-two pieces. I have two of those pieces. I have two-thirds. And I'm going to write that number in the numerator. The numerator tells you how many. The denominator tells you how big. The arithmetic of fractions is identical to the arithmetic of whole numbers. There is no difference between the two. If you actually understand what you're doing with whole numbers, you will immediately understand what you're doing with fractions. Well, that does require you to understand what you're doing with whole numbers. So here's a reminder. Everything you do in arithmetic is keeping track of how many of each type of unit where we do bundle and trading as necessary. So let's take a very simple example to get started with. Suppose I want to simplify this fractional amount five over three. Well, again, the denominator denotes tells us the size of each unit. And so here my denominator is three. And so that tells me my units are thirds. Now, if I want to make this problem as hard as possible, I'll try and figure out what this is from this point. But let's make our life easier and draw a picture of the thirds. Paper is cheap. Draw things, write things, scribble things out. Doesn't make a difference. Paper is cheap. Let's go ahead and do things the easy way. We'll draw a picture of a third. What's a third look like? Well, maybe it looks something like that. The actual appearance of the unit doesn't matter. Remember how you draw the units doesn't make a difference as long as you keep in mind what each unit is. And in particular, don't forget that bundle and trade rate. Because these represent thirds, because they represent thirds, because they represent thirds, what I have to remember is three of these things will form a next larger unit. Well, how many of them do I actually have? Well, the numerator tells us how many numbers the units. So here the numerator is 5. The numerator is 5. So that means I have 5 of these things. So let's go ahead and draw those. Here's 1, 2, 3, 4, 5. And there's a picture of what this fractional amount is. 5 thirds, there's 1, 2, 3, 4, 5 of these things. And each of these is representing one third. And again, the key observation here is we can always bundle and trade, because arithmetic is a matter of keeping track of what you have bundling and trading as necessary. So I have 5 of these thirds, but each of these units is supposed to be one third. So that means I can bundle three of them together to form a single unit. So let's bundle this set of three right here. I'll go ahead and paint that darker blue here. So these three together form a single unit. That's actually a 1. And so now I can indicate what we have. Arithmetic is bookkeeping. And so I want to indicate how many of each type of unit that I have. So first of all, the thing I notice here is I have 1 big blue thing. So my 5 thirds is 1. And then there's a couple of things left over. Well, again, arithmetic is bookkeeping. I have two, one, two of these things. And each of them represent a third. So I have two thirds left over. And two tells me the number. That's my numerator. Thirds names the size of the unit. That's my denominator. And so I'll write that as two thirds. And I have this form, which is the mixed number, one and two thirds.