 We can extend our fraction universe by talking about proper and improper fractions and mixed numbers. So it's a proper fraction. In the fraction A over B, the denominator B tells us how many pieces we've cut our unit into, and the numerator A tells us how many of those pieces we've taken. Since you can't take more than you have, and only a very selfish person would take everything, we expect A to be smaller than B. But what if it isn't? So for example, let's draw a picture of four fourths. What's it equivalent to? Now the denominator four tells us we've cut a cake into four equal pieces. The numerator four tells us that we've taken four of those four pieces. But that's the whole cake. This means that four fourths is one cake. And this suggests the following. For A not equal to zero, A eighths is equal to one. Now let's think about this a little bit more. In the fraction A over B, we've cut a cake into B pieces, and we take A of those pieces. Now mathematicians like to generalize ideas, and so we can generalize this fraction A over B if we cut a cake into B pieces, and then we take A of those pieces, or wait for it, pieces of the same size. So for example, suppose we want to draw a picture of three over one. What? Oh, right. Remember, how you speak influences how you think. We should always read fractions as A beats. So this is really three once. What's a once? It's what you get when you cut a cake into one piece. So I'll take my cake and I'm going to cut it into one piece. I don't actually cut it because it's already one piece. So in other words, it's a whole cake. So three once, we take a cake and cut it into one piece. In other words, we don't cut it at all. And then we take three pieces exactly the same size. But that's the same as three cakes. And so this suggests the following in general. For any A, A over one A once is equal to A. So now we can define an improper fraction. A proper fraction is a fraction where the numerator is strictly less than the denominator. Five eighths, three tenths, or this horrifying thing. An improper fraction has a numerator greater than or equal to the denominator. So nine eighths, twelve tenths, or this horrifying thing. The most important thing to remember about improper fractions, improper fractions are treated in exactly the same way as proper fractions. So let's draw a picture of five thirds. What is it equivalent to? So the thirds says we're going to cut a cake into three pieces and take five of those pieces. But one cake only gives us three thirds. And that means we need a few more pieces of the same size. So we need to cut up another cake. And so we see that five thirds is one cake and two thirds of another cake. So we can write this as five thirds is one plus two thirds, which gives us the sum of a proper fraction and a whole number. And this leads us to the next important concept. And that's the idea of a mixed number. A mixed number is the sum of a whole number and a proper fraction. We write it by juxtaposing the whole number and the proper fraction. In other words, we put them right next to each other. So this sum, one plus two thirds, we would write as one two thirds. And we typically read this as one and two thirds. So for example, let's try to reduce thirty-eight-twelfths to a mixed number. First, it's useful to remember that this is an improper fraction. Our numerator is larger than the denominator. And improper fractions are treated in exactly the same way as proper fractions. And what that means is that everything we can do with a proper fraction, we can also do to an improper fraction, and one of those things is to reduce the fraction. Now it's worth keeping in mind we don't actually have to do this. And big numbers shouldn't bother you, but it is easier to work with small numbers. So let's try to reduce this fraction by removing any common factors. And it's worth keeping in mind a factor is only relevant if it's a factor of both numerator and denominator. In this case we note that twelve has a lot of factors, but thirty-eight doesn't. Thirty-eight factors has two times nineteen, and there are no other factors. And so we look to see if twelve has a factor of two or nineteen. And it does. Twelve is two times six. We can remove that common factor and have our improper fraction in reduced form, nineteen-six. So how you speak influences how you think. So we should read this as nineteen-six. And we might make the following observations. Six-six is equal to one. Well that means twelve-six is equal to two. Eight-teen-six is equal to three. So now let's put it together. We have nineteen-six. That's eighteen-six and one more sixth. So we have our whole number part, three, and our fractional part, one-six. And so nineteen-six becomes a mixed number, three and one-six. Now it's worth making a comparison. Nineteen-six is equal to three and one-six. But if we read this fraction as nineteen divided by six and do the division, we get three remainder one. And this leads to the following general theorem about mixed numbers. Let a divided by b equal some quotient with remainder r. Then a-bethes is the mixed number q and r-bethes. And an important idea to keep in mind, don't memorize theorems. Understand concepts. In this case the crucial concept is that this improper fraction nineteen-six was eighteen-six, which is a whole number, and one-six left over. So for example, let's convert five and three-eight to an improper fraction. In truth there's no really good reason to do this, but we'll do it anyway. So our denominator eight tells us we're dealing with eights. And so the thing we might notice is that one is eight-eight. Two is sixteen-eights. Three is twenty-four-eights. Four is thirty-two-eights. And five is forty-eights. We also have three-eights. So five and three-eights is going to be forty and three-eights. Which we can write as the fraction forty-three-eights. And since there's no good reason to convert a mixed number to an improper fraction, it's not worth comparing five and three-eights equals forty-three-eights. And noting that five times eight plus three is forty-three. And this suggests a theorem not worth memorizing. You can convert the mixed number q and rb to the improper fraction q times b plus r over b. Again, don't memorize theorems. Understand concepts. And here the important concept is that five and three-eights will five is forty-eights. Three-eights is three more eights. And so all together we have forty-three-eights.