 Let's look at the addition using the place value chart in a second fashion. Part of the disadvantage to doing the addition the other way is that it requires a lot of drawing and redrawing. Now, if you're using actual objects, this is a little bit easier, but in practice we want to be a little bit more efficient. So let's take a look at the same problem we did the last time, 2, 3, 4, base 5, plus 3, 4, 2, base 5. And this time we'll do the addition still using the place value chart, but using our standard number symbols. Our convenience will also use our symbols 5, 6, 7, 8, 9, and possibly higher numbers. And the important thing to remember, these symbols don't actually exist in base 5, which is to say they're not really used and they don't have any meaning in base 5. They only have meaning because they are familiar to us. And so first off, I'll set down an empty place value chart. And again, I put down four columns because I feel like it. And again, for no particular reason, we don't really need them. They don't have any significance. But I'll set down the names of the units. So my right-most column, those are ones. My next column over, again, base 5 means that I can take 5 of these, and that's my next unit, 5 ones. I would call it 5. My next column over, 5 of these, 5 5s. Well, we can write that as a 25. Again, this is meaningless when I'm talking base 5. It only has meaning from an external context. And then 5 of these is a 125. And again, the thing to keep in mind here is that I don't really need to know the size of the unit. It doesn't really make a difference what these units are. The only thing that really matters is I'm working base 5, which is to say 5 of anything is 1 in the next place over. So let's go ahead and put the two numbers of our sum, 2 3 4 base 5 plus 3 4 2 base 5. And it's convenient to work from smallest place to largest place. So there's my two numbers. And I'm going to do what I call adding normally. But here's the important thing. I want to stay within each of the columns. Now, here's something that's worth noting. When we did this with the concrete representations, we were able to use the set definition of addition. We just ran things together. And the cardinality of our set is the sum of the cardinalities of the two. Now that I'm using the abstract symbols, I can't do that. I can't run 4 and 2 together and get 4 2 as my answer. I have to do something a little bit different here. And what I can do is, well, to some extent, I already know what the answer is. But what we're actually relying on at this point is the piano definition. If I add these two together, I get the second number after 4. So I can add, but at this point, I'm using the piano definition. So adding 2 plus 3 is 5. 3 plus 4 is a number that it is convenient to write down as 7. 4 plus 2 is a number that is convenient to write down as 6. Now here's something that's also useful to remember. Our goal in general is to write down things that are correct and complete. What does that mean? Well, at this point, my sum is correct. I have 5, 25s, I have 7, 5s, I have 6, 1s. So the sum is correct. It's not complete. I really want to write an answer in base 5, general rule of mathematics, and in life, give the answer in the same language it was asked in. So here the question was asked in base 5. I want to give an answer in base 5, and this 5, 7, 6 does not exist in base 5 because none of these symbols have meaning in base 5. So it's a correct answer. It's just not a complete answer. I need to do that final step. So the thing to remember is that because we're working in base 5, my bundling and trading says that I can take 5 of anything for 1 in the next place over. So let's take a look at this. So starting with the right again. So 6 is a bundle of 5 with one more. And so I can break that 6 apart. It's really a 5 and a 1. And this bundle of 5 I can trade for one more in the next place. Well, what do I have here? Well, I have a 7-1, well, that's conventionally, that's an 8. But again, I am working in base 5, so I look for those bundles of 5. So an 8 is a 5 and a 3, and I have a bundle of 5 which I can trade for 1 in the next place over. And I can combine these two, 5 and 1 is a 6. And again, I look for my bundles of 5, which is a 6 is a 5 and a 1. I didn't really need to include the 5 that I didn't really need to bundle the first time, but I did it just for consistency. And again, I can trade 5 here is 1 in the next place over. And so there's my final answer, 1, 1, 3, 1 in base 5, and I can read it off.