 So, an important idea is the concept of the order of operations. So, let's introduce some terminology. There are three important types of symbols in a mathematical expressions. An operator indicates something is to be done. These are things like plus, minus, divide, times. An operand is what the something will be done too. And these are things like the numbers 5, 17, 35, and so on. And finally, a relational symbol indicates a relationship between two expressions. The most common relational symbol that we have is equals. And it's very important to recognize that equals means that one thing can be replaced with another. So, for example, 5 plus 3, if we do the plus thing to 5 and 3, we get 8. And we might write 5 plus 3 equals 8. And a very important thing to keep in mind, when you write a equals b, you're guaranteeing the replaceability of one side with the other. You are guaranteeing that 5 plus 3 can be replaced with 8 anywhere we see it. And unless you're willing to stand by your guarantee, don't write the equals. 12 minus 4, again, if we do the minus thing to 12 and 4, we get 8. And we write 12 minus 4 equals 8, because we are willing to guarantee that wherever you see 12 minus 4, you can replace it with 8. 2 times 4 equals 8. If we do the times thing to 2 and 4, we get 8. And so we write 2 times 4 equals 8. 35 divided by 5. If we do the divide by thing to 35 and 5, we get 8. And so we write 35 divided by 5 equals 8. And we're willing to stand by that guarantee this is not actually willing to guarantee this. And so we could write not equal by writing a slash through that equals. And upon further consideration, we might be willing to guarantee that 35 divided by 5 can be replaced with 7. So we have to ask ourselves, self, what happens if an expression has more than one operator? We must agree on the order in which we'll perform the operations. Otherwise, different people will evaluate the expression differently. Now, order of operations is like deciding on which side of the road we're going to drive on. We could drive on the right side of the road or the left side of the road. And as long as we all agree on which side of the road we're going to drive on, it's not a problem. The difficulty is when some people decide to drive on the right, at the same time other people are driving on the left. And so we have to agree on how we'll perform the operations. And the general rule for order of operations is this. Perform operations from left to right unless... So for example, say we want to evaluate 12 minus 7 minus 3. The basic rule of order of operations is we'll go left to right unless... And at this point we haven't hit the unless. So we want to evaluate from left to right. So the first thing we find is 12 minus 7. And that's equal to 5. And equals means replaceable, so we could replace 12 minus 7 with 5. And remember, if you didn't change it, it's still there. We only replace the 12 minus 7. The minus 3 is still there. Now the next thing we see is 5 minus 3. So we find 5 minus 3. Equals means replaceable, so we could replace 5 minus 3 with 2. And there's nothing left to do at this point. So again, the basic rule of order of operations is we perform operations from left to right unless... And the most important unless is a grouping symbol. These are things like parentheses where left parenthesis begins a group and right parenthesis ends it, brackets, braces, and others. But we try to avoid these in higher mathematics because they tend to have more specialized meanings. There are also what are called implied grouping symbols. And we'll talk about those a little bit later. And the important unless here is that operations within a set of grouping symbols are always done first. So let's evaluate this horrible mess. And we see there is a set of parentheses. And the grouping symbol means we have to find what's inside first. But wait, there's a grouping symbol here too. So we have to find the stuff inside the parentheses 5 plus 1 first. And so we find 5 plus 1 is equal to 6. And so we replace 5 plus 1 with 6 everywhere we see it. And remember, if you didn't change it, it's still there. So we still have the 12 minus parentheses 7 minus parentheses 6 parentheses parentheses. Now we still have this grouping symbol. So that means we have to find parentheses 6 first. Since there's no operator, we do nothing. Parenthesis 6 is just equal to 6. And we can replace parenthesis 6 with 6 everywhere we see it. But wait, there's a grouping symbol here. Parenthesis 7 minus 6. And so that means we have to evaluate 7 minus 6 first. Equals means replaceable. So every place we see 7 minus 6, we replace it with 1 everywhere. Again, the parentheses require us to evaluate parenthesis 1 first. But since there's no operator, parenthesis 1 is just equal to 1. And equals means replaceable. Every place we see parenthesis 1, we'll replace it with 1. And finally we can evaluate 12 minus 1, which is 11. And equals means replaceable. So every place we see 12 minus 1, we'll replace it with 11. Now there are a few situations where a grouping symbol is implied. It's not written down, but it is there. And the most common place we'll see this is as the numerator or denominator of a fraction. There are also a few other places where we have implied grouping symbols, but we'll talk about those later. So if I want to evaluate 4 plus 3 over 8 minus 1, because this is the numerator and denominator of a fraction, there is an implied grouping symbol. And a useful thing to remember is that the numerator and denominator of a fraction should always be enclosed in parentheses. Well, paper is cheap, so let's rewrite our fraction where we've enclosed the numerator and denominator inside a set of parentheses. And our order of operation says we have to do things inside parentheses first. So we'll perform the operations inside the parentheses first. That's 4 plus 3. And so every place we see a 4 plus 3, we'll replace it with 7. This 4 minus 1 is also inside a set of parentheses, so we'll evaluate that. Equals means replaceable, so everywhere we see 4 minus 1, we'll replace it with 3. And since there's nothing more to do in parentheses 7 and parentheses 3, well, parentheses 7 is just 7, parentheses 3 is just 3, and equals means replaceable, so we can replace and get our final answer.