 Now the set definition of addition allows me to construct a different algorithm for addition and this is going to be derived as follows. So let's go ahead and try an addition using a place value chart and we'll use a concrete representation of our objects and let's add together 2, 3, 4, base 5 with 3, 4, 2, base 5. Now in this problem I want to use a place value chart with concrete representations. What do I mean by that? Well we'll set down an empty place value chart for base 5 where each unit is 5 of the preceding unit and we know that we're dealing in base 5 because our numbers are in base 5. So there's our place value chart, a place, a place, a place, a place and I've written down 4 places here I could have more. I've written down what each of the units look like although I don't really need to. Again here's my 1, 5 of these form the next unit, 5 of these form the next unit, 5 of these form the next unit and where I artistically inclined 5 of these would form the next unit and so on. Again I don't actually need to draw what the units look like, they're just there because I happen to have some free time and I'd like to draw. So let's see. I'll go ahead and set down my numbers. My numbers are 2, 3, 4, base 5 and 3, 4, 2, base 5 so I'll set those down and I want a concrete representation. These are abstract symbols that tell me how many of each of these I have so let's go ahead and write down a concrete representation and I'll do that by replacing our abstract number symbols with concrete representations of what they are. So here I have 4 of these things so rather than writing 4 I'll write down a bunch, 4 of these little squares. I have 3 of these and I have 2 of these. Likewise I have 2 of these, 4 of these, 3 of these and there's my concrete representation of the sum 2, 3, 4, base 5 plus 3, 4, 2, base 5 and well now I can use the set definition. I have 2 sets, I'm going to put them together into a single set and that's my sum and so I have a set of these, I have another set of these, I run them all together and there's my sum 2, 3, 4, base 5 plus 3, 4, 2, base 5. Well I'm not quite done yet, I actually want to find what that sum is so I'll go ahead and do the bundling and training. Remember I'm working in base 5 which means that any time I get 5 of something I will be able to bundle it and trade it into the next column. Now here's a rule for cleaning, work in one direction and in this particular case I want to work from smallest to least because what's going to happen I'm going to bundle some stuff and maybe move stuff into here. Bundle and move over. If I work from right to left I never have to go backwards. The problem is if I bundle here and move over I might bundle here and drop some things into here that may cause me to have to re-sweep. So let's sweep in one direction, working from the smallest place, I'll take a look at this and look I have a bundle of 5 here that I can put together and well that doesn't belong here anymore, it really belongs over here so I'll slide it over. And again I have a bundle of 5 here and again this doesn't really belong here, it belongs to the next place over and again I have a bundle of 5 here and again this doesn't really belong here it belongs to the next place over and I have no more bundles I can make so now I'm ready to write my final answer and again what I'm going to do returning to my abstract symbols I'm just going to record how many of each of these that I have. So I have 1, 1, 3 and 1 and so my final answer is 1, 1, 3, 1 and reminder we're working in base 5 I'll write that down at the bottom.