Hi everyone and welcome to our first lecture, the lesson on real numbers. I know we're all here for algebra and to get better basic math, but it's really important that we start off with what goes in to these variables, what goes into these functions and that of course is real number. In my experience, people come from different backgrounds with this, they have different definitions, they have different terminology. So it's extremely important that we all sort of start off on the same page. And one thing I want to show you is that the real numbers, often if I ask people to draw this and represent this, they draw a number line, and that is perfectly fine for most purposes. You put zero in the middle and you have infinity off on the right and negative infinity on the left. But I think this diagram, this picture hides so much information and really doesn't get it to the subtle nuances of what different kinds of numbers are out there. So I'd like to sort of walk through and talk about different kind of numbers, so treat these as almost definitions. You don't need to memorize these right now but keep them handy, I'd recommend taking notes as you go, for new definitions in particular, always think about referencing them later and also examples of things that are and are not those terms. So let's start off first with something called the natural numbers. Here's my exercise to you, pause the video and count to 10. I am very serious, pause the video, I'll wait count to 10. Even if you want to say it out loud, here's my guess, no matter where you are in the world, you started at 1 2 3, I have a small son, he can count to 10 skips 7 but we're working on that. So he starts at 1, so there's no real reason to start at 1, would you be wrong if you said 0 1 2 3 or 7 8 9 10? I didn't really specify a starting place, but it's a very natural place to start and go sort of 1 2 3 4 and count your way up. These numbers are called the natural numbers. If you think about it, and we're counting with our fingers, of course we started 1 2 and 3, if we were cavemen and we're counting how many will the mammoth we killed this day and we take our rock, and we etch it into the cave, we go 1 2 3, this is a naturally great place to start. So these natural numbers, 1 2 3 4 5, we denote this by an N, with a double bar on the left. Now, fast forward from caveman time to sort of India each in India, and think about nothing, the next number that entered the number system was zero. Now this is not an obvious jump and it took a very long time to get there, is a little difficult to symbolize, formalize and conceptualize nothing. So we have 0 1 2 3 4 and off we go, when you add zero you have created the set of whole numbers. I remember whole because of the O in whole and of course zero looks like an O, so we have the set of whole numbers. That set is noted as W with a slash to denote the set of whole numbers. Now fast forward throughout human history and start trading with different civilizations and all of a sudden, you owe me money or coins or whatever it is back then, rocks that they traded, who knows? So if you owe me something, you can have a debt and after zero came negative number. Now there's no limit to how much you can be in debt, so we go dot dot dot, -3 -2, -1 0 1 2 3, you can certainly have profits. And now we have sorted to start to form the larger system of numbers that we know as integers. So this set of numbers is called integers. Now be careful here for different reasons. We don't denote this set with the letter I we denote it as Z. If you speak a little German, if you Sprinkle as a little Deutsche, sorry I don't but one of the ways to say number in German is zahlan, Z-A-H-L-A-N. So the Z, comes from German for zahlan. A lot of math, the first math textbooks, a lot of universities that have a lot of mathematicians were in Germany and France. And so you will often see a lot of German influence, into the symbols that are in or the language or descriptors vocabulary of math. So here's the first one, Z for zahlan. We don't use I, we tend to save I for interval, a little interval around the number, so I does get used but it's not here, so Z integers and this is Z. If I want to talk about a particular element, I'm going to use some symbols here by the way. So I want to say 1 is an integer, but I wanted to remove the English sentence, look and feel of this and I wanted to write this in symbols, I'd use the symbol that looks like a little E, but it's not an E, I'd say one is an element of Z. one is a set a member of the set of Z. This symbol you think of this as an element of, and certainly you can write things that are not an element of. So I could say for example that zero is not a natural number, and you start to get into more of the symbolism of math than writing out full sentences, which is a bit more common now after integers. Fast forward to Egypt or Greece are some ancient civilization and start thinking about architecture and beautiful buildings and ratios, and you get yourself into the set of rational numbers. Rational numbers are ratios or sometimes called fractions of integers, I write it is A over B and then I write a couple of things. This bar by the way means, such that, so the way that we have this so far is the set of all numbers A over B, such that A and B are integers. I'm allowed to pick any integer I want, two thirds, one half, three fifth, pick whatever you want as long as of course, and I think we all know this, you can't divide by zero, so you're not allowed to pick your denominator to be zero. Any other values are fine, as long as your denominator is not zero. These are called irrational numbers. The symbol to denote these ratios is not R, why can't we use R? We're going to save R for the real numbers, so I have to come up with another clever way to say ratios or fractions, I can't use F because I'm going to use F for functions. So I look up in other words in the thesaurus and I come up with Q for quotients. When you write a numbers one third or two thirds, remember you're dividing one divided by 3, 2 divided by three, and so you get quotients. So Q is our set of rational numbers. It's a very famous result that not every number is rational. You may know that pi cannot be written as a fraction. So we'd say of course that like pi is not rational. When you are not rational, some like this, you're called irrational, so pi is irrational. E square to 2, there's all these numbers that cannot be written as fractions. Once you have all your rational numbers there, when you start to want to talk about the more complicated numbers, pi, square to 2, E, pick your favorite scary number, then that is when you get to the real numbers. So the real numbers is our big world or universe, is where we going to hang out for the most part. But I want to show you the structure that happens inside of this universe that's not obvious from the number line. Real numbers is a little difficult to define, if you ever take a more advanced course called real analysis, so do this officially. I'm going to sort of hand wave a little bit and say any number with a decimal expansion that allows me to talk about 7, right? because 7 of course is like 7.0000 or I can certainly talk about pi, which is 3.14159 dot dot dot. So every thing is a real number. I want you to stare at this for a second, realized that as I've developed a larger set, I haven't lost any numbers along the way. This is extremely important, I want to keep all my previous contributions to the number system. So when I went from natural to whole, it's zero plus everything I had. When I went from the whole two integers, its negatives plus everything I had. Rationals, I can think of every integer as a rational, how do I do that? How do I think of 2 as a fraction? Well that's just 2 over 1. How do I think of a rational number is a real number? Well I would take out the calculator and figure out its decimal representation, so one half is point five, one third is point 3 repeating. So I can always always always, you going to sue me I'm not dividing by zero, turning a rational number into a real number and you get everything along the way. What I like about the number line is that it's short, it's sweet, it's simple to think about and it's definitely useful at times, but I don't think it gives the true picture. Let me draw you a different world at different universe of numbers, just so you can get a feel of the structure of the real numbers here. We said the very first level, the simplest level are the natural numbers. And inside the natural numbers we have to remember we start at 1 2 3, when I add zero I get this extra layer which are my whole numbers. When I start to add negatives I get another layer which is my set of integers. Keep going and form fractions or ratios or quotients of integers, and you get rational numbers. Finally add numbers like pi and E or any other decimal expansion is not a fraction, and you get the set of real numbers. And these delineations, these lines, these separations are nontrivial, there is a reason why we draw the line. So for example, what is a whole number that's not natural? What some number that lives in the second layer? Well zero, what's an integer that's not whole? So over here we have -1, -2, these are different layers where different numbers live. What's a fraction quotient that is not an integer? How about two thirds? How about negative 5 halves to make this? Of course there infinitely many choices and fractions like a pick. What's the real number? That is not a fraction pi is a famous example, how about route 2? How about E? Have you seen me before, this is like 2.718 this is whole number it's on the calculator, somewhere. There are lots and lots more the so there are definitely levels and layers to the structure of real numbers that are certainly not obvious by just drawing the number line. If you want to write this and set notation, we tend to write this if you want to be a little more formal about it using the subset notation, so we say N is a subset of W. Natural numbers are a subset of the whole W is the subset of Z, Z is a subset of Q, Q is a subset of R. This is the world, this is the universe that we will be focusing on in this class, this is what I want you to know. So if I start talking about injures, I want you to have in mind what they are and what they are not. I talked about rational numbers what I want to show you why scored the two is not rational. I just want you to have in mind what these definitions are and the symbols that are used to talk about these sets or subsets of numbers. Now, I'm not supposed to tell you this of course but there's more out there, and maybe you know this but this is a big secret. We don't really talk about them in this class but there's one more layer. At least one more layer we could draw. Okay, you in theory keep going but the next layer out would be your complex numbers. I'm just going to mention this here is in case you've seen these before, but these are complex numbers. This is the set of numbers like A plus BI, and this is I the imaginary numbers. So sometimes complex or imaginary numbers are called the same. A and B, you can pick any real number you want, and I, here's the thing that I think maybe you've seen this, maybe not. If you haven't seen this, ignore everything I'm saying right now but I is a square to negative one in particular I squared is negative one. So it's some imaginary number that when I square it gives me negative one. This notion of squaring something to get a negative, you can't do that with real numbers, so there's no number, no real number that if I square it gives me negative. So the next layer out if you did want to keep going becomes your complex numbers, these are your complex numbers. That's the note with a C with a bar on it, and if you've seen complex numbers they look like 3 plus 2I or you can get 7 plus 0I of course this is just 7. So you still get everything you have pi plus 0I's, just pi. So you don't lose anything prior you just get the ability to get new numbers. Now I's in there 3I 7I whatever you want them to get a whole another layer of numbers. We won't talk too much about complex numbers in this class, but it's great if you know what they are and if you take more advanced classes, you will certainly see them. Okay, keep this sheet handy, we're going to build off of this great job in the first lecture and we'll see you next time.
