 ratios, ratios, ratios. Now before we get into unit conversions, we're gonna have to take a look at ratios and figure out how to deal with them, okay? And for us to be able to do that, we're gonna have to take a step back all the way to series one and take a look at the real number set and see how ratios fits into the real number set. As we talked about before, that's basically the basis of what we're started with and we're gonna continuously go back to the real number set. And the real number set, as you recall, it was talking about the natural numbers which was, you know, just the counting numbers going 1, 2, 3, all the way up to infinity counting forever and ever and ever. And then from there, we were able to define the number zero and by being able to do that, we created a whole new category called the whole number set, right? Which is basically took the natural numbers and included zero and called that it's sub, you know, another set which is the whole numbers, right? From there, we moved on to the integers which was basically using zero as a pivot and reflecting the natural numbers into the negative realm. From there, we went on to the rational numbers which was basically any numbers that we can express as a fraction of integers. And from there, we went into the irrational numbers which was basically any numbers that we couldn't express as a fraction of integers. And that whole group, the natural numbers, whole numbers, integers, rational numbers and irrational numbers, we put into one huge category called the real number set, okay? So we're going to have to take a step back to the real number set and see how ratios fit into the real number set, okay? So basically what we have is the real number set here. We have the natural numbers, the whole numbers, the integers, rational numbers and the irrational numbers, okay? Now, as you recall, this half of the real number set is the rational numbers, right? And rational numbers are any numbers that we can express as a fraction of integers. And that is where ratios kicks in. So rational numbers, we have it as a fraction of integers where we have one number on top of another number where the denominator cannot equal zero. So the first way we started talking about these numbers was, when we first started defining rational numbers, we call them a fraction of integers, right? And fraction has a specific meaning on its own, right? Let's say, for example, we had the number 2 divided by 7. The way we say this, the way we say this in English, the way we represent this in English is 2 out of 7. So what that means is there's a total number of 7 objects and we're taking 2 out of them, right? And that is what fractions means. When we're writing fractions, when we specifically in mathematics say, hey, we're writing a fraction out of something, we're talking about taking something out of a whole, okay? And that's what fractions is, 2 divided by 7. So what we're stating in here when we write this as a fraction, what we're doing is saying when we write 2 over 7, what we're saying is 2 out of 7, right? So if you had a total of 7 boxes, 2 of them would be green. So when we write 2 out of 7, what we're really doing is saying there's 2 green boxes out of a total of 7 boxes. Now, I colored these as orange, the remaining boxes as being orange, but we're not really defining the remaining boxes as being anything, right? The total, the 7 represents the total number of boxes. I just colored them green, but we don't know anything about those boxes other than that, that they're boxes, right? This 7 down here doesn't mean that they're 7 orange boxes, it means that they're 7 boxes total, right? So when we say fractions, what we're doing is we're saying there's 2 of something, if you write 2 over 7, 2 of something out of 7 total, right? We're talking about a certain number out of a whole. And this stuff kicks us really into one of the, it's one of the first steps we do when we take, when we go into percentages and when we go into probability and statistics. So when we start talking about fractions, when we start exposing ourselves to fractions and we start talking about parts of a whole, this kicks us into percentages, which is sort of the first step that we take to go into probability and statistics, okay? And that, all of that connects all the way back to rational numbers and the real number set, right? And it's basically our way of sort of using a fraction of integers in the real world, wherever we want to use it, right? And that's one definition of what a rational number is, right? It's one definition of what one integer above another integer means, which is 2 out of a whole, something out of a whole. And that's what fractions is and that's main, the main way we've been using what rational numbers are. Now there's another way which fine tunes it, gives it a sort of a distinct different meaning when we write one integer on top of another integer. And that is ratios, okay? And that is what we need to learn to get into to start doing unit conversions, because when we write one number on top of another number, then we have to sort of sit back and decide what that means, right? When we say it's a fraction, we're saying it's part of a whole, it's 2 out of 7, something out of a whole, right? But there's another meaning, there's different terms that we have in English, right? What if we wanted to say it's 2, 4, 7? What if we wanted to say it's 2, 2, 7, right? Something for something else. And that gives a totally different meaning and that's how we deal with ratios. So the way we define ratios is, is not 2 out of something, right? We define it as being 2 for something. It's something for something else, okay? It's comparing two numbers of the same type, of the same kind. And this is where we're going to go. This is what we need to learn to be able to do our unit conversions. So for ratios, when we write 2 over 7, so for ratios, when we write 2 over 7, the meaning has slightly changed. Over here, when we said 2 over 7 for fractions, we had a total of 7 objects. Over here, for ratios, when we write 2 over 7, or yeah, 2 over 7, the meaning changes and now we have a total of nine objects because there's nine things in play. We're putting two orange boxes in for seven orange boxes. So what happens here when we write 2 over 7, when we're talking about ratios, this becomes 2, 4, 7. 2, 2, 7. Two green boxes for seven orange boxes. And there's a little bit of more meaning behind this because now we know what the other boxes are. We know specifically that they're actually seven orange boxes. When we're talking about fractions, we didn't know what the other boxes, what the color of the other boxes were going to be, right? I just put them as orange because I had the color orange, right? Over here, when we write it down as ratios, when we write down seven orange boxes, it really means seven orange boxes. And the total of boxes in play here is actually nine. When we write it down as a fraction, 2 over 7, there's a total of seven boxes in play here. When we write it down as a ratio, when we say it's a ratio of 2 over 7, there's a total of nine boxes in play here. And this comes into play, the most basic place that it comes into play is when you're dealing with, you know, if you're gambling, if you're doing trade, right? I'm going to give you two dollars for seven apples, right? However way you want to think about it, right? If you're looking at a map, there's two centimeters equals seven kilometers, right? And that's where the unit conversion comes in. And that is the main difference between fractions and ratios. So over here, ratios, this is going to directly kick us into unit conversion, which we talked about, we did an introduction for in video number 45 or 46, okay? Now, the beauty of all this is the difference between fractions and ratios, when we write two integers like this, if we say it's a fraction, the meaning is this. If we say it's a ratio, the meaning is this, right? Now, the meaning changes slightly, but our rules and mathematics do not change. The axioms that we've developed that we've learned when it comes to addition, subtraction, multiplication, division, when it comes to the rules of dealing with the equal sign, when it comes to rules of dealing with exponents, right? The rules of mathematics, our axioms do not change. The meaning of the numbers changes. So when the rules of mathematics do not change, our definition of the numbers change, then we don't have to learn new rules to be able to deal with ratios. We don't have to learn new rules to be able to deal with fractions. The rules are the same. So everything that we've learned previously applies both to fractions and ratios. Nothing changes based on the rules of mathematics, based on the axioms we've learned. The only thing that has changed is the meaning between what we've written down. And that is the beauty of mathematics because it's a universal law. All it is, as I've stated before, math is a very personal language. It really depends on how you're expressing yourself, what your intentions are, what your meaning behind numbers are that you're writing down. If you write something down and say it's a fraction, then you're talking about a part of a whole. If you write something down and say it's a ratio, then you're talking about comparing two different numbers. One thing to keep in mind, we talked about rational numbers and we said rational numbers are any numbers that we can express as a fraction of integers. When it comes to ratios, the numbers do not necessarily have to be integers. We can write one number on top of another number and say it's a ratio and it still makes sense. If you go to the store, you can give a dollar and buy one and a half apples. You can go pay 50 cents and buy half a watermelon if there's a place where you can pay 50 cents and buy half a watermelon. So keep in mind that ratios do not necessarily have to be integers. Ratios, all they are is one number on top of another number. What you're doing is you're comparing one number to another number of the same type during a transaction, of the same dimensions. It may be just currency. Whatever it is, you're comparing two numbers. May they be things, may they be ideas, may they be concepts, may they be powers and gains, may it be whatever it is. All you're doing when it comes to ratios, you're comparing one number to another number and what that means is there's something here and there's something there and it's going into a pool if you're talking about putting things into a pool or if you're talking about converting, you're converting one thing to another. And that's what ratios are and all of this probability statistics unit conversion, they all come back to here which kicks us back into rational numbers. And that's why we started off with the real number set because the real number set is the basis of us laying down the foundation of what we understand numbers to mean. And if you think this is brilliant, this is cool, wait until we get into the irrational numbers because the irrational numbers are infinitely larger than the rational numbers. And if you think this is interesting, then irrational numbers, once we get in there, that'll blow your mind. And that's basically what ratios are. What we're going to do from here is talk about a little bit more and take our two systems that we've talked about, the color squares and the color triangles and look at the connection, look at the link, look at the anchor between them and from there start doing conversions between one system to another system. So we become familiar how to do the conversions and there's a couple different conversion formats that we're going to talk about.