 So, now we need to introduce the idea of order of operations, and this is sort of like deciding which side of the road we're going to drive on. We have to agree. There's no inherent advantage of one order or the other, but we do have to make an agreement. Now, because addition is associative and commutative, the order of addition isn't important. So, if I have an addition, 5 plus 7 plus 3 plus 2, then the order that I do the addition doesn't matter. So, maybe I'll add 5 plus 7 together, that gets me 12, and then I'll add 3 plus 2, that'll get me 5, and then I'll add 12 and 5, that gives me 17. And here I'm using my associativity of addition to take any two terms I want to add together. And because addition is associative and commutative, it doesn't make a difference, so maybe I'll add 7 plus 3 together and get 10, and then maybe I'll add 10 and 5 to get 15 and 17. And so, no matter how I add, I get the same answer. But as soon as I have a second operation, the order in which I perform these operations can become important. So we have to agree on an order of operations. So for example, 5 minus 2 plus 1, well, if I do 5 minus 2 first, I get 3, I add 1, and then I get an answer of 4. But if I decide, well, let me do this 2 plus 1 first, see what happens then. That gets me 3, 2 plus 1 is 3, and then 5 minus 3 is 2, and I get two different answers depending on what order I do those operations on. Because we don't want to have one expression having two possible answers, we have to agree on an order of operations. And again, it's like deciding which side of the road we're going to drive on. It doesn't matter as long as we agree. So the fundamental rule about order of operations is the following. We're going to do everything. All operations are going to proceed from left to right. And again, you can think about this as the rules of the road. Here's how we normally do things. We normally drive on the right unless you're told otherwise. And what we'll tell you otherwise are a couple of different things. The first important exception is you have an operation within a grouping symbol. These grouping symbols are things like parentheses, braces, brackets, and there's a couple of other grouping symbols that we'll introduce along the way. If you see any of these things, braces, parentheses, brackets, they give you an unless. Things within those grouping symbols should be done first. They have, to use our driving analogy, they have the right-of-way. The other possibility is you have operations with a higher precedence. So these have to be done in precedence order. Again, if you want to use the driving analogy, you can drive along the road, but there are vehicles with higher precedence. Any sort of emergency vehicle with its sliced flashing says you have to give way. And this is the same idea with order of operations. Now addition and subtraction are equi-precedent. Neither one goes before the other. And so our default rule, do all operations from left to right, is the one that holds. And again, the only other possibility is that there may be an unless there, which is to say in this particular case, the grouping symbols. So here's another important idea. Now that we have many different operations or two different operations, an arithmetic expression is identified by the last operation that's performed. So if I take this operation 5 minus 3 plus 7, there's no grouping symbols. Addition and subtraction are equi-precedent, so I go from left to right. The last thing I'm going to do is add. So this is a sum. Again, I go from left to right here. The last thing I do is that subtraction. So this thing is a subtraction. And I go from, oh, here's my unless. There's a parentheses here, so I do stuff inside the parentheses first. The last thing I do is add. So again, we have a sum. Well, let's try that. Let's place parentheses so that we can make our statement true 8 minus 5 plus 2 minus 1 equals 0, and then we should identify the type of expression. And so, well, maybe there's no parentheses at all. And because addition and subtraction are equi-precedent, I'm going to evaluate them from left to right. So that says I take care of 8 minus 5 first. That's 3. 3 plus 2 is 5. 5 minus 1 is 4. And that's not equal to 0. So I know there have to be parentheses in here if the statement is actually going to be true. So let's think about that. Well, the first thought may be to put parentheses around the 8 minus 5. But because I'm evaluating things from left to right, if I throw parentheses around 8 minus 5, that says to evaluate that first. But I'd evaluate that first in any case. So there's no point in putting the parentheses there. So maybe I'll throw them around the 2 minus 1. So let's see what happens. I throw parentheses around 2 minus 1. And parentheses say do stuff inside first. So that gives me 8 minus 5. And my do stuff inside, that's going to be 1. And again, I have to now, since I have subtraction and addition, they're equi-precedent, I go from left to right. So the first thing I evaluate, 8 minus 5. That's 3 plus 1 is 4. Isn't 0. Isn't what I want it to be. So where else can I throw the parentheses around? Well, maybe I'll throw them around the 5 plus 2. So let's take a look at that parentheses. They do stuff inside first. So that's 8 minus 7 minus 1. And going from left to right, 8 minus 7 is 1. 1 minus 1 is 0. And that's what I want it to be. And so now let's take a look at this expression. The last thing I actually evaluated was a subtraction. So the type of expression is going to be a subtraction.