 So, why does this method work? Well, it probably looked pretty straightforward to you. We were rewriting our number in a format that we knew how to do the arithmetic for. But there's a few things that probably went by pretty quickly. If I start with a number 4 on 2, this time I'm in base 5, and now I want to convert to something in base 10. Now I've got something a little more complicated. I can rewrite this in terms of base 10. So here I've got 3 times my base is 5 here. Really, my base is 10. My base is always going to be 10 because each of these places represents 10. 10 is going to be different in different bases, but it's always just 10. Now the exponent here is 0, 1, 2, 3. And I have 4 times 10 squared plus 1 times 10 to the first plus 2 times 10 to the 0. So the first thing I really need to do is make sure all of these numbers are in terms of my destination base. If they're not, then I can't do the arithmetic on them. So first of all, all of these coefficients are small. My source base is already smaller than my destination base, so I'm not going to be worried that one of these numbers is larger than my destination base, and I might have to do some conversion there. On the other hand, my bases are different. So 10 here is different from 10 over here. So 10 in base 5 is really 5 in base 10. So I'm going to want to convert all of my bases into my destination base. So that will give me 3 times 5 to the third plus 4 times 5 squared plus 1 times 5 to the first plus 2 times 5 to the 0. Now I've got something that's all in decimal, and this I can do arithmetic on. The things up here were intermediate stages where each piece of this needed to be converted, until it was in decimal, and I could do the arithmetic on it. I could be more explicit and go back and add that all of these are in base 5 and just needed to be converted to base 10 before I could do the arithmetic. But here we've got nice simple conversions we can do. Our bases are usually easy because chances are we've written them down in decimal to begin with. Other numbers are often easy, but if you've got something that's larger than your destination base, say if we are converting from hexadecimal to decimal, then we'd want to convert those into decimal as well before trying to do our arithmetic.