 So any mathematics above the level of counting is going to require you to have to deal with fractions. Fractions emerges follows. It's often convenient to deal with part of a whole. So we might have one slice of cake instead of a whole cake, part of a class of people instead of all of them, some money out of a pile of cash instead of all of it. In these cases we are breaking up a whole into fragments, and so we invent these broken numbers to represent these fragments. And that's where fractions come from. We take a whole, break it up into pieces, and assign a fractional value to each one of the fragments. Well, a picture is worth a thousand words, and it's always time for cake. So suppose we cut a cake into four pieces. Four equal pieces. Now each of the piece is a fraction of the cake, and so we say that each piece is a unit fraction. And since there are four equal pieces, each piece is a fourth. Let's introduce some terminology. The denominator of a fraction tells us how many equal pieces we've split our unit. So if we've cut our cake into three pieces, each piece is a third. Or if we cut the cake into six equal pieces, each piece is a sixth. Or maybe we've cut our cake into four equal pieces, and in that case each piece is a fourth. The numerator of the fraction tells us how many of the equal pieces we're taking. And so we'll express the fraction as the numerator, the number of pieces, over the denominator, the size of the pieces. And here's an important idea. How you speak influences how you think. So if we see a fraction like this, we should read this as three, that's the number of pieces, fourths, which is the size of each piece, and we should not read this as three over four. So let's show three fourths of the cake. Since the denominator is four, the cake has been cut into four equal pieces. Since the numerator is three, we take three of those pieces and let's box them up to go. An important idea is the notion of an equivalent fraction. If we cut a slice of cake into several pieces, but take all of the pieces, we obtain an equivalent amount. It's the same amount of cake, it's just the pieces are smaller. So here we've cut our cake into four equal pieces, so each piece is one fourth. Now if we make some additional cuts, there are now 12 equal pieces. If I want to have the same size cake as before, I need to take three of these twelfths, so I can take three twelfths or one fourth, and it's the same amount of cake, so I can say that three twelfths is equal to one fourth. And since these are equal, we say they are equivalent fractions that represent the same amount. So for example, I might try to find two fractions equivalent to two-thirds, and so we can start by sketching two-thirds. The denominator three tells us that we've broken the unit into three pieces. The numerator two tells us we've taken two of these pieces. Now if I want to form an equivalent fraction, we can cut all of our pieces into smaller amounts. So if we cut every piece in half, then the original was cut into six pieces, so each piece is a sixth. But if I want to get the same amount of cake, we need to take four of these pieces, and so the amount represents the fraction four sixth. Or maybe we can cut all the pieces into three parts, and so there are now nine total pieces, so each is a ninth. And if I don't want to short change myself on cake, we need to take six of these pieces, and so we need six ninths. So notice a couple of things. We found that two-thirds is the same as four-six, where we had two times as many pieces, but we had to have two times as many to get the same amount. We also found two-thirds was equal to six-ninths, where we had three times as many pieces, but to have to have three times as many to get the same amount, and so this leads to the following theorem. Now remember, we should read these fractions as two-thirds, not as two over three, six-ninths, not as six over nine. Unfortunately, that construction does get a little bit awkward, so we are going to revert to our usual way of talking about fractions as something over something. And so we'll say that in general, for any fraction A over B, and any non-zero number N, A over B is equal to N A over N B. So for example, let's find three different fractions equivalent to five over twelve, what? Oh, right, five-twelfths. So our theorem says that if we multiply numerator and denominator by the same number, we get an equivalent fraction. And so we can pick any non-zero number N, so we'll pick, well, how about N equals two, and if we do that, five-twelfths is two times five over two times twelve. And if we multiply these numbers out, that's ten-twenty-fourths. We can pick any non-zero number we'd like to, so how about let's pick N equals six. And so five-twelfths is six times five over six times twelve, and that's thirty-seventy seconds. We need three different fractions, so we've picked N equals two, N equals six, so let's pick N equals, well, how about N equals three? So five over twelve is three times five over three times twelve, or fifteen-thirty-sixths.