 So now we're going to start looking at how we can convert between bases. In the previous video we saw a couple of bases that are of special interest to us. Binary, decimal, octal, and hexadecimal. But first we're going to look at four different ways to convert between any two arbitrary bases. So this will allow us to convert between, say, decimal and binary, or decimal and base 13, any two bases that we're interested in. And then we'll also look at a special purpose way that will allow us to convert between specific bases. In this case things like binary and octal or binary and hexadecimal. Now we're going to be looking at four different ways to convert between arbitrary bases and I generally recommend that students learn at least a pair of them. The reason is that each of these methods requires you to do your arithmetic in either the source base or the destination base. If you pick a pair of them, where one of them allows you to do your arithmetic in the source base and the other in the destination base, then you'll be able to choose which one you want to do your arithmetic in based on the problem that you've got. All four of these methods have the same basic logic behind them. And we'll look at how each of these is derived from basic concepts. But they're fundamentally looking at what is a number. What does this thing mean for us? So given a number like, we'll be able to say what is this number precisely? What does this number actually mean? So first of all, we typically write a subscript with the base at the end. This allows the reader to easily see what base we're working in so that we don't get this confused with say hexadecimal or even octal here. The one exception is hexadecimal where we'll occasionally write a 0x at the beginning to indicate that it's a hexadecimal number followed by whatever digits I've got. So looking at this number, I can actually expand this to tell you really what this number signifies. And the way I'm going to do that is by adding in all of the information about the places. So with Arabic numerals, all of these places have some implicit meaning. And for the moment, we're going to make those explicit. So I'm going to expand this number using scientific notation, allowing me to explicitly represent all the information about each of the places. So I'll start by working from left to right. I have a 1 in my first position. And this is times 10 to the 0, 1, 2, 3, 4. And then I'll look at my second position. So I've got 3 times 10 to the 0, 1, 2, 3rd position. And then I have 6 times 10 squared plus 2 times 10 to the first. Plus 5 times 10 to the 0. This expression is exactly the same thing as this. But now I've made everything very explicit. And it's going to be a whole lot easier to convert between this expression and a corresponding expression in a different base. I can, of course, repeat the same process for, say, beef over here. Where I have b times 16 to the third. Plus e times 16 squared plus e times 16 to the first. Plus f times 16 to the 0. This time the big change is that I've got a different base here. Before I was working in base 10, now I'm working in hexadecimal. So I have 16s for all of the bases. Each of these places represents 16 possible values. So moving over each place, we have 16. We have 16 options in each of these cases. So we're working in base 16. And accordingly, each of these is being multiplied by 16. Over here we have 10 possible options. So we're multiplying by 10 each time we're moving over one place. When we convert between two different bases, the basic idea is that we want to find the coefficients in the new base that correspond to the expression that we had before. And that's going to be what underlies all of the different methods that we used to convert between two arbitrary bases.