 So, this time we're going to start looking at numbers. And we're really going to begin at the very beginning and build our way up. So, probably the first thing to talk about is, what is a number? Why do we have them? What do they help us with? So, a number is a way of representing the concept of a certain quantity. And we can represent these with symbols. We can represent them with graphs. And humans have come up with lots of different ways to represent numbers over the last few millennia. But when we think about, why did we develop the ones we did? Why did we keep the ones that we have now? We're going to have to think about, what do we do with numbers? Do we really just represent quantities? Or do we actually do something with them? Well, we commonly like to do arithmetic on them. We like to perform operations that increase or decrease the quantity we have. We'd like the representation for our numbers to be able to help us with this. So, each of the ways we've come up with has different advantages and different disadvantages. Some of them help you in certain ways, make it easy to read your number. Some of them make it easier to do computation. But all of them fundamentally mean the same thing. So, here I've got a table that's got lots and lots of numbers here. Got numbers all the way up to 100,000. And they've got all of the standard number systems that you're probably used to seeing. You're hopefully used to seeing Arabic numerals on a daily basis. You may have seen Roman numerals before. They're still used occasionally. And if you're familiar with East Asian languages, you've probably seen common system used there. Each of these, again, as I said, has different advantages and disadvantages. So, the advantage to things like Roman numerals in the Chinese, Japanese, and Korean system is that the numbers actually give you some idea of the magnitude of the entire number. So, if I'm looking at large Roman numerals, you'll notice I have some really odd symbols. You start getting m for 1000, c for 100, this weird combination of c's and d's inside themselves. And all of that gives us some really nice idea about how large our number is. And the same thing happens with the Chinese, Japanese, Korean system. I have a symbol for 10,000. I have another symbol for 1000. And these systems are, to some degree, combinatorial. If you look at Roman numerals, you'll notice we're kind of adding things together. I have lots of i's. And if I have two i's, well, i plus i is two. So that's what I'd expect. Then I have three i's. But then they also have subtractions. So, I have v, which is five. And I'm subtracting one because my i is before the v. So, I've got five minus one is four. Then I go back to having five plus one is six. Five plus one plus one is seven. Then I've got ten minus one. So, I'm getting an idea about how large my number is. But this isn't a terribly convenient system to work with. This would be even harder if I had to try to do arithmetic with it. Sure, some of it would be nice because sometimes you've got, say, minus ones and plus ones. And those would cancel out. But otherwise, you're left with things like, I've got, say, five plus one plus one plus one and five plus one plus one. And now I'm having to add up all of these terms. I've got two fives, five ones. Together that's 15, but I'm going to have to do a whole lot more work to get there. The Chinese-Japanese-Korean system kind of sits between the Roman numerals and Arabic numerals. In that, yes, we have these magnitude terms, like you can see the ten there. And it's getting duplicated in all of these. I've got eleven, twelve, thirteen, fourteen. In the meantime, I'm just taking the regular symbol one and copying it after the ten. And if I go farther down, I see, okay, I've got two times ten plus seven. So that gives me twenty-seven. On the other hand, that also means I've got twice as many symbols to write. There is some benefit that I don't have any placeholders the way I do with Arabic numerals. So I have forty here, so I have four and a zero. Over here, I just say I have four tens. So that can be really, really nice when you get really large numbers, because I don't have to read all of these zeros and count five zeros and try to figure out where I am in this number. Instead, I can just say I have ten, ten thousands, which will get me to understanding my number a little bit faster. But it still may not be ideal for performing arithmetic. I've still got lots more symbols to work with, and some of them aren't actually productive towards my arithmetic. The way you're probably used to doing arithmetic involves lining up numbers, so that numbers that have similar magnitude are all in the same column. And that would just involve extra symbols here that aren't really helping us. But they do tell us something about our magnitude. What we're going to do, though, is we're going to try to extend our basic Arabic numeral system to support other representations. And our computers are designed in such a way that they can, that they really only understand two things. They understand ones and zeros. And even those are an abstraction off of the actual hardware. So it will be relatively simple to produce a computer that can do binary arithmetic, but it would be a lot harder to build a system that can perform decimal arithmetic. So in our binary number system, we've got the same idea. We've got a number line here. It starts at zero, counts up by one at a time. But this time I only have two symbols to work with. I have zero and I have one. So once I run out of ones in my number, I have to increase my next binary place. So after one comes ten. I don't have a two. I can't go to two. I have to go to ten. I'm increasing my tens position and resetting my ones position to zero. Then when I add another one, so ten plus one gives me eleven. Because I can add one more in that ones position. Zero plus one gives me one. And I'm still just holding the tens position constant. But then if I want to add another one, well, eleven plus one has to be a hundred. Because I've got no more room for anything in the tens or the ones position. So I have to carry over to the one hundred position. Octo and hexadecimal are the same sort of idea. But this time we have a different number of symbols to count with. In octo I have eight symbols to count with. In hexadecimal I have sixteen. Obviously I don't have sixteen symbols that I commonly use to represent numbers. So we had to come up with something else to use after we run out of the first ten that we are used to working with. So in this case we've just taken the first six letters from the Latin alphabet. So we expect that A in hexadecimal means the same thing as ten in Arabic. Or E in hexadecimal represents fourteen. We'll also see that there's some really nice correlation between binary, octo and hexadecimal. Because their bases are exponents of binary. Binary has a base of two, octo has a base of eight, which is two cubed. And hexadecimal has a base of sixteen, which is two to the fourth.