 Okay, so let's take a look at some important properties of addition. So consider an addition expression like 1,998 plus 147 plus 2. Because we read from left to right and because addition is a binary operation, we have no choice. We have to interpret this as the sum 1,998 plus 147. Now we can actually do this. We can find 2,145, then add 2 to get our final answer, 2147. And if we had no capacity for independent thought, like if we were a computer or a calculator or a political commentator, we'd have to perform addition this way. On the other hand, if we are capable of independent thought, we can look for and very often find a better way. Consider the rationale for looking very closely at what we're doing in basic arithmetic, and many other things, is that if you think about what you're doing, the natural question is, is there a better way of doing this? So let's take a closer look. Well consider the commutative property of addition. Well, we're going to consider a sum like a plus b. And again, since addition is defined in terms of the addition of two sets, and since we read from left to right, what this tells us is that we're going to take a set of a, and we're going to drop down and add to that a set corresponding to b. And I'm going to put those together. And this combined set here is going to represent my sum a plus b. On the other hand, sometimes I start with a set corresponding to b, and then I drop down a set corresponding to a. So again, here, I started with my set b. I joined to it my set a. And so now I have this combined set that corresponds to the addition b plus a. B was here first. I've added a to it, and here's my combined set b plus a. And the thing to notice is that it's the same set in either case, whether I started with a and added b to it, or whether I started with b and added a to it. I got the same sum. And so that tells me that there's something about addition. It doesn't matter which way I do it. And we talk about this as the commutative property of addition. So if I have two whole numbers, a and b, then a plus b, set a to which set b is added, is the same as b plus a. Set b to which the set a is added. And so addition is commutative. It doesn't make a difference what order we write it in. Well, there's another useful property. And again, this goes back to our first example. Addition is a binary operation, and we can only ever add two numbers at a time. So if I consider a sum like a plus b plus c, again, we have to read this left to right. This says take a and add b to it. So I'll set down a, I'll add join to set b. So here's my representation a plus b. And then I'm going to then to that set add c. So there's a plus b plus c, and there's my sum. On the other hand, what if I set down to set b first? Then join to it a set c, and then added my set a. So this is b plus c. I did that first, and then I added a. And as before, we get the same sum even though we added differently. And again, this is an example of a very useful property called the associated property of addition. And again, in general, if I have three whole numbers a, b, and c, then what I get by adding a plus b plus c first is the same as what I get if I were to add a and b together and then add c. I can associate any two terms in the addition. Well, these are great and useful theorems, but they're generally useless. In general, memorizing a theorem is a waste of time. What's important is to be able to use the theorems effectively. Also, the other reason that theorems are useful is, again, the theorem itself might not be all that useful. Using it is important. And what you go through to establish the theorem is useful because it reminds you of things that you should know or that you have learned. So the associative and communicative properties of addition together means that we can add in any order that we want to. So let's take a look at that addition that we started with. 1,998 plus 147 plus 2. So if you were a calculator, a computer, or something other, something incapable of independent thought, you would add these two together, then add two. On the other hand, because you are capable of independent thought, you can decide, well, I want to add 147 and 2 first, because that's kind of easy. And actually, if I think a little bit ahead, I'm not going to add those two quite yet because what I want to do is I want to reverse the order of that addition. And for the reasons for that, it'll become apparent in a second. I can reverse the order because commutativity says I get the same thing. 147 plus 2 is the same as 2 plus 147. Well, I get to apply associativity again. I don't have to add these two. I can add these two. Why would I ever want to do that? Well, part of the reason I might want to do that is that's a much easier addition. 1,998 plus 2 is 2,000 plus 147. And I can do that addition in my head. In fact, if you look at this, we don't really have to write all of this out. If we understand how to use the associative and commutative properties of addition effectively, then we know that we can reorder any addition in any way that we want to. So given something like this, for a minute, think about how you might do this. Well, again, I could plot long 1,998 plus 147. And I have this standard addition algorithm. I could spend some time working at that and getting the wrong answer. But what I'll probably do is look, 1,998 plus 2 is 2,000 and pick up the one I forgot. 2,000 plus 147 gives me this. If you know how to use the associative and commutative properties effectively, you can do this addition in your head. Well, let's take a slightly easier one. For example, 8 plus 7 plus 5 plus 3 plus 6 plus 2. And if I'm a calculator or computer or otherwise incapable of thinking, that's how I could do this. On the other hand, it's worth noting that it's easy to add 10 to a number. So what I might want to do is identify pairs of numbers in here that add to 10 and use those as the basis for my addition. This is a standard technique that's taught everywhere in the world except for the most part in the United States, and it's what's known as the adds to 10 approach. And so we take a look at our number, and we think about what we're doing and say, ah, I know that 8 and 2 are a nice pair that add to 10. And let's see. Oh, I know 7 and 3, also a nice pair that add to 10. And well, 5 and 6, I don't add to 10, but they are the last two numbers left and I could do that addition. That's not too difficult. That's 11. And so what do I have? I have 10, 20, 11, 31 as my partial sum. And again, in practice, if you were to do this type of problem mentally, they think you wouldn't even bother drawing all these. These are just illustrating what you're actually doing. What you're doing in practice, 8 and 2 is 10, 7 and 3 is 10. These are left over. That's 10, 20, plus 11, 31 is my sum.