 Hi, this is Ciccio, end of May 2009, and welcome to series 3 of the Language of Mathematics. Now what we're going to talk about in this series is going to be from what I figured out to be the most important symbol in the language of math, which is the equal sign. But before we get into understanding the equal sign, which is really directly related to units, units are basically specifying what it is that we're talking about in math. But before we get into that stuff, we have to do a little recap of what we've covered so far, which is basically dealing with the real number set, dealing with operations, addition, subtraction, multiplication, division, so basically crunching numbers. And what we did was because we're going to get into stuff as fast as we can, we also introduced exponents, which was in series 2, with radicals. Exponents of really radicals are the same thing, just fractions and not fractions. If you need a recap, if you don't know what I'm talking about, take a look at the videos. What we're going to do right now is go through this stuff really quick and then start talking about how to release an equal sign. So let's go to a little wall here and just do a little quick drawing sketches and visually try to understand or remember what it is that we talked about before and how it applied everywhere. If you remember the first stuff we got into when we started talking about the language of mathematics was the real number set. So let's do a little sketch of it and talk about its subset and just go through it really quick. Talked about this before and basically this is the real number set and what we have on this side is the rational numbers and on this side are the irrational numbers. Now rational numbers were broken down into four categories. The natural numbers were the counting numbers, one, two, three all the way up to infinity. You had the whole numbers which was the introduction of the number zero and then you had the integers which was positive and negative whole numbers and then you had rational numbers which were fractions of integers. Anything that you could write as a fraction of these guys is a rational number. On this side you had the irrational numbers and the irrational numbers are anything that you cannot write as a fraction of integers. Integers or natural numbers are really prime numbers and again we went into a lot of detail in this in the first series so if you need a recap of prime numbers, take a look at the prime numbers. So with the real number set, if you break it down to its basics this side is anything that could be expressed as a fraction of primes as number one or as number zero and this side is anything that cannot be expressed as a fraction of prime numbers and that was the real number set. From here we went and dealt with, we learned how to deal with the numbers and from there we introduced operations. So we took numbers and said what can we do with these numbers? Well we added numbers, we subtracted numbers, we multiplied numbers, we divided numbers so from here we came and we added the plus sign, the subtraction sign, the multiplication sign the division sign which is basically the four ways that we deal with the real numbers. So there's a whole bunch of videos that I've already done which deals with this and deals with operations and then what we decided to do in series two was introduce another layer to our numbers basically kick it up to another level. So initially we had, for example, we're going two times, this is a show up, let's do green two times five would have been ten, right? So this was our regular operation with the real number set taking our rational numbers these are just integers or natural numbers, right? You could go negative two here, all of a sudden it becomes an integer, right? So we went two times five is equal to negative ten, you got negative times positive, negative times positive is negative. What we did, we introduced exponents. So we took our base numbers here, right? And we said, hey, what happens if you put a number here? What does that mean in the language of mathematics? What does that symbol do? So if we put a two here, all of a sudden we had exponents and what we did with exponents was, hey, we created rules to represent or to deal with the symbols that we've created. So we took our operations really and applied them to exponents. So in multiplication, these two numbers multiply together, it gives you ten. If you had two things multiply here with the same base and could do different exponents, those guys add it. So series two deals with this stuff. Series one deals with the real number set and the operations. And it also has a section on trigonometry and geometry which we're going to talk about more again in every series. It's just going to grow and grow, right? So we're going to take all this information from series one and two and apply it somewhere. And as soon as we apply it somewhere, we're talking about the equal sign. And when we're applying it, we have to know what we're talking about and that implies specifying what units we're talking about. So we're going to take this information, apply it somewhere which is basically dealing with the equal sign and when we're applying it, we're going to assign units to our numbers, right? So we're going to try to see if one thing equals another thing, right? I think this comes up everywhere, sustainability when you're talking about the environment and equality if you're talking about human rights in business. Liabilities, assets, you have to balance the budget, right? What's going to happen is we're going to take things and try to see if they're equal to each other. So for example, let's do the real number set, right? When we did the series and we draw the real number set or when I draw the real number set, I break this up into two separate halves, right? Equal halves. But in reality, there's a lot more irrational numbers than there are rational numbers. So visually, this is really good just to remember how this is laid out. But in reality, there's way more irrational numbers than there are rational numbers. So there are areas, and we've talked about this in German, if we haven't talked about it, we will talk about it. The areas of these two things may be equal visually, but the number of numbers we have here is way less than the number of numbers we have here. So these two sides in reality are not equal when it comes to quantity anyway when we're talking about quantity. Visually, yeah, the two areas we draw is equal areas, right? So again, there are two areas that might be the same, but the quantity in each section is going to be different. So again, we're going to take these rules and apply them to places and learn how to, again, crunch numbers that have units in them and try to balance our equations. And then we're going to talk about getting into functions. This goes directly into functions and trigonometry, right? So we did graph, x, y, axis, and we're going to graph things and follow them up as functions. So series one, we're learning about the real number set, trigonometry, geometry. Series two, we learned about exponents and how to deal with our operations and exponents. And series three, we're going to take all everything that we learned in series one and series two and apply it and see where we go with it. And this might be a continuity because we can apply mathematics, or anyone can talk about mathematics, use mathematics to talk about anything.