 Alright, so let's start a look at addition in base 10, and one of the things that the place value chart and trading and bundling leads us to is a method that's called totals below. And so previously, we've added by setting down a place value chart, writing down our add ends, and then adding within the places, bundle, trade, and combine, and writing our final answer. All addition algorithms are based on these steps. Every addition algorithm is one variety or another of these steps, and the only significant difference is how many of the steps we don't show. So for example, let's consider the addition 287 plus 5394. Now since no base is specified, we can assume these numbers are written in base 10. And as before, we'll set down our place value chart, but again, the unit names don't really matter, so we'll omit the unit names, but we will include the add ends. So there's our place value chart. I haven't bothered writing the unit names, but my add ends 287, 5394, and I'll add within each place, and this time, again, for organizational purposes, I'll write each total on its own line. So I can add left to right. It doesn't matter. I'm adding within each column. I can add left to right. I can add right to left. I can skip around and add whichever columns I want to. I'll add left to right. So 5, 2 plus 3 is 5, 8 plus 9 is 17, 7 plus 4 is 11. Now the next step is going to bundle sets of 10 and trade. And again, here, I do want to go from left to right, not because it's inscribed on stone tablets handed down from ancient times, but because if I don't go right to left, then I may have to retrace some of my steps. So it's convenient to go right to left. I could go left to right if I wanted to, but if I do, then I might have to do a little bit more work. So let's see. This 11 is a set of 10 and 1. So that's going to be one more in the next place over. This combines. This 17 is a set of 10 and 7 more. So I'll trade the 10 for 1 in the next place over. And there's nothing else I could bundle. And now I'll do my combining 5, 5 plus 1 is 6, 7 plus 1 is 8, and 1 is 1. And there's my answer. And if I want to show this using the totals below, the only thing I'm going to change is I'm going to omit the place value chart and there's my answer. Now the important thing to notice here is it doesn't make a difference whether we add left to right, right to left, or completely at random, as long as we add everything. So again here's my addition to 87 plus 5,394. And this time for variety's sake, let's go ahead and add from, oh I don't know, right to left. So I'm going to add 7 plus 4 is 11, 8 plus 9 is 17, 2 plus 3 is 5, 5 is just 5. And now I'm going to add 5, 6, 8, and 1. And there's my answer. Now I'll make a couple of comments here. We have a standard algorithm for addition that's been around for a couple hundred years. And if you take the standard algorithm apart, the only difference between the standard algorithm for addition and what we've just done here is rather than writing out everything neatly on its own line, what we've done is we've smashed all these lines together, gotten our answer in one incomprehensible mess. And so the difference between the standard algorithm for addition and this total's below method is we've just made our work a lot more comprehensible. Now what's given that, why do we use the standard algorithm? Well there's one reason why the standard algorithm is popular. It saves space. If you use the standard algorithm, you don't have to write these four lines. You save space. Now if paper were a couple hundred dollars a sheet, there may be some value in saving space. But paper is cheap. Thought is expensive. Understanding is expensive. It doesn't do you any good to save cheap paper at the cost of understanding and at the cost of comprehensibility, which is a key thing to keep in mind. Because one of the things we want to be able to do is we want to be able to understand what we're doing and we don't want to make things more difficult. We want to make things more comprehensible and more understandable.