 Our familiar number system is base 10, so in contrast to addition in base 4, the 10 for one trade is harder to draw. Now we'll begin using our abstract numbers symbols to represent the amounts, but it's important to remember that they represent the number of units, so we'll continue for now to use the place value chart. Now because our number system is base 10, we actually have names for the higher units, and we typically say that the smallest unit is a 1, 10 1's make a 10, 10 10's make a 100, 10 100's make a thousand, and if we're writing a number in base 10, we'll omit the base designator. So what we used to write as 1 3 5 base 10, we'll write as 1 3 5 without indicating the base. Moreover, we'll read the number using the unit names, and so this number should be 100 3 10's 5. What's that? What do you mean 30? So let's talk about the names of the numbers. We also have the familiar names for the numbers 1 2 3 4 5 6 7 8 9 10, and the number after 10 should be called 10 and 1. However, in defiance of logic and sensibility, we call it 11, and so this brings up an important idea. Always distinguish between the easy and the familiar. So it might seem strange and unnecessarily difficult to call the number after 10 10 and 1. Why not call it 11? That's easier. Well, actually it isn't. It's familiar. And so the thing to remember is that every new number word is something that must be learned and memorized. So calling the number after 10 11 requires remembering a new word. It also loses the connection to the fact that it is the number after 10. Remember how you speak influences how you think. In contrast, calling it 10 and 1 is both logical and retains the connection. So if we counted logically, we'd count 10 and 1, 10 and 2, 10 and 3, 10 and 4, 10 and 5, 10 and 6, 10 and 7, 10 and 8, 10 and 9, 10 and 10. Now we can make one further modification. This last one, 10 and 10, would actually make more sense as two 10s. And again, this might seem unnecessarily difficult. Why not call it 11, 12, 13 and so on? And again, always distinguish between easy and familiar. It's familiar that we count 11, 12, 13, but it's not easy. We have to remember all of these new names. In contrast, if we count like this, we don't have to remember any new names and we retain the connection. And in fact, by way of comparison, let's consider the following. In Chinese, the numbers from 1 through 10 are, in my mangled Mandarin, yi, ar, san, xi, wu, liu, ji, ba, ji, xi. The number after xi, 10, is xi, yi, 10 and 1. Then comes xi, ar, xi, san and so on, up to xi, jiao, 10 and 9, which is followed by ar, xi, two 10s. And what this means is that once you can count to 10 in Chinese, you can count to 20. In contrast, in English, you need to know 20 different words to count to 20. Actually, it gets better. Once you know these 10 words and the naming convention, you can count as high as 9, 10s and 9, what we might call 99. And again, in comparison, in English, you need 27 different words to count that far. And Korean, Japanese and several other languages have a similar structure to their number words, making the numbering much easier. Now remember how you speak influences how you think. And if we read numbers this way, this will help us think about numbers in a useful manner. So let's read the numbers by specifying the number of each type of unit. So this number might be called 12, but we don't want to read it that way because 12 doesn't tell us anything useful. This is actually one 10 and two 1s. Similarly, this number, two 10s and four 1s. And this is one 10 and one 1. So why is it useful to read numbers this way? Well, suppose we wanted to add these three numbers together. If we read these as 12, 24 and 11, it's not really clear how we can add them. But if we read them as what they are, this is one 10 and two 1s. This is two 10s and four 1s. And this is one 10 and one 1. And remember arithmetic is bookkeeping. So if we add them by combining the sets, we see that all together we have four 10s and seven 1s, which we can write this way. And that gives us our addition. And what's important here is you can actually do this mentally. One 10, two 1s, two 10s, four 1s, one 10, one 1. That's one, two and one 4 10s and two, four and one 7 1s. Now we might call this mental arithmetic, but what we're really doing is we're talking our way through to the answer. So we might add these numbers mentally. This number is 100 and three 1s. This number is two 10s and one 1. And this number is one 10 and three 1s. Arithmetic is bookkeeping. We set it all together. We have 100, three 10s, and seven 1s, which we can write this way.