 We can use the idea that the natural numbers form an ordinal set, that is to say every natural number has a 1 after it, and we can improve it through the use of what's called benchmark counting. And the basic idea is this, we can count up by 1s, but we don't have to, we can count up by larger amounts. So rather than just using the arrow to indicate the number after, we can indicate the 5th number after, the 10th number after. And we have to modify our arrow notation to keep track of these larger amounts. A lot of mathematics is bookkeeping, and how we can do that most efficiently. So for example 5 plus 3, well I could view that as the 3rd number after 5, and I might count 6, 7, 8, but if I know the 3rd number after 5, I can just jump right to it. Now to indicate that I have actually jumped to the 3rd number after 5, I might superscript that arrow with a plus 3. And the important idea here is this makes the connection between the sequence of counting, the number after, the number after, the number after, and the idea of addition as indicating the nth number after a given value. One of the useful things that we can do to make this much more efficient is to consider how we can get to the nearest 10. For example, if I'm going from 6 to 10, well if I'm at 6, 10 is how many numbers after that, and I know that 10 is the 4th number after 6, or equivalently I might know the addition 6 plus 4 is equal to 10. And another way we might say this is we might say that 4 completes 6 to 10, which is to say 4 more to 6 will make 10. And so for example I might have a number like 17, I want to complete it to the nearest 10, and so the nearest 10 after 17 is going to be 20, and so I know I'm going to go from 17 to 20. And if I count up from 17, then I find that 20 is the 3rd number after, so I'll indicate that using my arrow diagram. Again, equivalently we might just remember that 17 plus 3 is what I need to make 20. Now a number like 20, or other numbers like 400 or 8000, or any number in general that ends in one or more zeros, we refer to these numbers as benchmark numbers, and they're actually very convenient numbers to focus on. And the reason is that it's useful to arrive at a benchmark number because counting from a benchmark number is generally going to be much easier than counting through a benchmark number. For example, if I consider the addition 17 plus 6, well that's the 6th number after 17, in other words it's 17 plus 6, and I can do the addition, it's going to be 23. On the other hand, what I might do is I might complete 17 to the nearest 10, and completing 17 to the nearest 10 puts us at 20. That's the 3rd number after 17. Now since I actually want to go to the 6th number after 17, I have to go a little bit farther. Well how much farther? Well I have to go 3 more, and so here's the thing that's worth noting. It is easy to go from here to here, because that's completing 17 to the nearest 10, and then it's easy to go from the benchmark number 23 more to 23, and all together I've gotten a total of 3 and 3, I've gotten a total of 6 to get to 23. Now the usefulness of this we can look at by something like finding the 35th number after 27. So this is the same as 27 plus 35, and you can think about how you might do that using the standard arithmetic algorithm, but here's a better way of thinking about it. So what I'm really trying to do is find starting at 27, the 35th number after 27, and what I might do to begin with is the first easy number I get to, the first benchmark number I get to is going to be 30, and that's 3 more after 27. Now what happens after that? Well after that, since I'm at a 10, it's actually pretty easy to go up by 10. Now I want to go up by 35 numbers, so I am going to go up by 10, and again, and again, so all together I've gotten up by 10, 20, 33. I want to go up to the 35th number, so that's going to take me 2 more to 62, and this tells me that the 35th number after 27 is 62, and this is the same as finding the sum 27 plus 35 equals 62. And we can turn this into a formal method of addition, which is based ultimately on counting.