 Now we're going to introduce a new type of number, which is our rational number and our fractions. And these emerge by a consideration of algebra. So if I take two algebra problems, 5x equals 17, and 3x is equal to 11, if I only have whole number arithmetic operations, I might try to solve two problems in the following way, 5x equals 17. And so what that means is that 5 times x is 17. So I can find x by dividing 17 by 5. And using whole number division, I find that quotient, that's 3, with a remainder of 2. All right, not a problem. But I can take 3x equals to 11, and I can try and solve it the same way. That's 11 divided by 3. And so x is also 3, remainder 2. And because x is 3, remainder 2 here, and x is 3, remainder 2 here, then the x's here seem to represent the same thing. However, there's a problem here, because if they represent the same thing, then 5x minus 5 of these x, minus 3 of these x, should be 17 minus 11. So 5x minus 3x, 17 minus 11, so 2x is 6. And that tells me 2 times x is 6, so x must be 3. Except it isn't. 5 times 3 is not 17. 3 times 3 is not 11. So there's something strange going on that we have to reconcile. And the problem is that we need a new type of number to represent what we obtain when we divide and have some sort of a remainder, so that when we divide 17 by 5, we don't get something that is what we get when we divide 11 by 3. We want to make sure that 17 divided by 5 is not 11 divided by 3. And so this requires us to introduce rational numbers. Suppose I take two whole numbers, a and b, where b is not equal to 0. And I can think about the quotient, a divided by b, because I can write that quotient in whole numbers. And again, b can't be 0. I can identify this quotient with the rational number, a over b. Now, it's possible that this quotient may be a whole number. So we'd like to make sure that our definitions overlap. So if a divided by b is a is equal to c, exactly with no remainder, then I'm going to define my rational number a over b to be that whole number. But if there is a remainder, then the rational number is just whatever it's going to be. So if there is a remainder, a divided by b is just the rational number a over b. Now, it's useful to have a good model for the rational number. And this emerges fairly easily. We have a model for division. And since the rational a over b corresponds to a division, we can use that model for division to produce a corresponding model for a rational number. So for example, let's consider the rational number 2 over 5. Now, the first thing to remember is that this corresponds to the quotient 2 divided by 5, which we can view as 2. So I'll put down a 2. So there is a 2. I'm going to divide this into five equal parts. So we're looking at this from a partitive viewpoint. So there's my division into five equal parts. And again, when you divide, so here's your cake, you're going to divide that into five equal pieces. And each person is going to get one of those equal pieces. So our quotient is just going to be one of those pieces, and we'll shade that. So we've taken our cake, our 2. It's been divided into five pieces. And each person gets one of those pieces. And there's our partitive division. So here's a slightly different way we can look at this problem. The important thing here is to recognize that this blue section here is 2 divided by 5. So I don't have to actually start with my 2. I can actually start with a 1, and again, divide that into five parts. Now, this time I want to get something this big from these parts. And so I'm going to take two of those parts. So here, this portion here is the same as 2 divided by 5. So this portion here, two parts out of the 5 of that we've divided the one into, is going to be our quotient 2 divided by 5, and is going to be our rational number 2 over 5. So here's a model for that rational number. So let's introduce a few terms here. In general, if I take a rational number p over q, I can represent this as taking a 1, dividing it into q parts, and take p of those to represent my rational numbers. So for example, if I want to make a model for 3 fourths, I'll take a 1. There it is. I'll divide it into four parts. And I'll take three of those parts corresponding to the rational number 3 over 4. Now, it's important to note that the two numbers that I've expressed here as the rational number 3 over 4 have two very different purposes. The lower number 4 tells you how many parts you're going to divide the 1 into. So here, the 1 has been divided into 1, 2, 3, 4 parts. And that's because our lower number there is a 4. And so what that lower number tells us is the size of each one of those parts. Now, if I take a look at just one of those parts, I can call that a unit fraction. And in this case, this is the unit fraction 1 over 4. Again, 4 is how many parts we've divided it into. And the unit says that this is how we're going to count. Our count will be of things that look like this. So now consider the upper number of our rational 3 over 4. So 4, the lower number, tells us that we've divided our 1 into 4 pieces. And what does that upper number do? Well, that upper number tells us how many of the pieces we're going to take to represent our rational. So there's 1, 2, 3 pieces. We're going to take three of those unit fractions. And so again, this gives us our model for the rational number 3 over 4. And again, our lower number identifies the type of rational number. It names it. So we call it the denominator. So nominate comes from you're going to name somebody for a position. And this is the lower number. So that's where the D prefix comes from. So this is the denominator. It's the thing that names it that is below. And the upper number 3 tells us the number of parts we have. 1, 2, 3 parts. And so we're going to call that the numerator because it numbers the parts. So our rational number has numerator 3. That tells us how many parts we have. Denominator 4, that tells us how big each part is. And altogether, one way we can look at this, because we have broken a part of 1, we have broken it into fragments, we can call this rational number 3 over 4 as a fraction, which again comes from the word to break apart.