 Alright, now that we have a good idea of what subtraction really is and how we can do this without having any sort of formal algorithm for it, let's see if we can develop a formal algorithm. And for that we want to go back and start with our concrete representation of numbers using our place value chart. And so let's consider a problem like 432 base 5 minus 231 base 5. And so I can set down my place value chart. Again, I don't really need to show what the units look like. It's just a convenient visual picture there. However, the thing to remember is since we're working base 5, each unit is going to be 5 of the previous unit. So 5 of these make this, 5 of these make this, 5 of these make that, and so on. And the numbers that I have here are going to be 432 base 5. Let's see, that's 4 of these, 3 of these, 2 of those. And I want to remove 231 base 5, 2 of these, 3 of these, 1 of these. So let's take a look at that. So I'm going to set down my 432 base 5. And because I want to work with the concrete representation, I know I have 4 of these, 3 of these, 2 of these, but let's go ahead and draw those in. So there's my 4, 3, and 2. And then I want to remove 231, 2 of these, 3 of these, 1 of these. So I'll pick out 2 of these to get rid of, and they're gone. I want to take 3 of these, well I'll pick all 3 of these. I want to remove these 3, and they're gone. And then finally, 1 of these, so I'll pick 1 and get rid of it. And so this is what I have left. And now I want to change back and record using my abstract symbols. I have 2 of these, 0 of these, 1 of these. So my remainder is going to be 201 base 5. Well, what if I try something else? How about 1012 base 4 minus 323 base 4. And so again, I'll set down an empty place value chart just to be able to organize everything. And again, I don't really need to do much more than remember that any one of these is 4 of these and so on. I don't have to show what those look like, it's just convenient to do so. And so I'm going to have my 1012 base 4, that's 4 digits, so that's going to be 1 of these, 0 of these, 1 of these, 2 of these. And from that, I want to remove 323 base 4, that's 3 of these, 2 of these, and 3 of these. So I'm going to remove that. And, well, I have a problem. I don't have 3 of these. I don't have 2 of these, and I don't have 3 of these. I can't remove this from here. I can't remove this from here or this from here. Fortunately, I can borrow, well, actually I don't want to use that term anymore. You might have been told to borrow, but the reality is you're never going to return what you borrow. It's gone, so the proper term is we're going to trade, and we're going to use the fact that our trade rate we're working base 4 is 4 for 1. So any one of these I trade, I'm going to get 4 of the next thing over. So I can trade any unit for 4 of the next. So I'll trade this for 4 of these, and this is gone now. No longer exist. And now I'm in a position to remove 3, because I have 3 I can take away. But I still have the problem over in this column. So I'm going to, again, pick one of these to trade for 4 more in the next place. So I'll trade. This is gone, and now I have 4 of these. And I have the one that I had originally. Keep track of that one. And we're good. Now I'm able to remove 2 of these because I have enough. But now I still have this problem at the last column. So again, I'm going to trade one of these for 4 more in the next place over. And again, this thing that I've traded, that's gone. No longer exist. And I can do the required removals. So I'm going to remove 3 from here, and that leaves me with nothing. I'm going to remove 2 from here, and then I'm going to remove 3 from here. And now I am in a position that I can actually write my answer down using my abstract number symbols. I have 2 of these and 3 of these. So I can write my answer 2, 3, and all together that difference 101, 2 minus 3, 2, 3, base 4 is going to be 2, 3, base 4.