Order of operations
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In this video we're going to talk a little bit about order of operations. And I want you to pay close attention because really everything else that you're going to do in mathematics is going to be based on you having a solid grounding in order of operations. So what do we even mean when we say order of operations? So let me give you an example. The whole point is so that we have one way to interpret a mathematical statement. So let's say I have the mathematical statement 7 plus 3 times 5. Now if we didn't all agree on order of operations, there would be two ways of interpreting this statement. You could just read it left to right, so you could say well, let me just take 7 plus 3, you could say 7 plus 3 and then multiply that times 5. And 7 plus 3 is 10, and then you multiply that by 5. 10 times 5, it would get you 50. So that's one way you would interpret it if we didn't agree on an order of operations. Maybe it's a natural way. You just go left to right. Another way you could interpret it you say, I like to do multiplication before I do addition. So you might interpret it as -- I'll try to color code it -- 7 plus -- and you do the 3 times 5 first. 7 plus 3 times 5, which would be 7 plus 3 times 5 is 15, and 7 plus 15 is 22. So notice, we interpreted this statement in two different ways. This was just straight left to right doing addition then the multiplication. This way we did the multiplication first then the addition, we got two different answers, and that's just not cool in mathematics. If this was part of some effort to send something to the moon because two people interpreted it a different way or another one computer interpreted one way and another computer interpreted it another way, the satellite might go to mars. So this is just completely unacceptable, and that's why we have to have an agreed upon order of operations. An agreed upon way to interpret this statement. So the agreed upon order of operations is to do parentheses first -- let me write it over here -- then do exponents. If you don't know what exponents are don't worry about it right now. In this video we're not going to have any exponents in our examples, so you don't really have to worry about them for this video. Then you do multiplication -- I'll just right mult, short for multiplication -- then you do multiplication and division next, they kind of have the same level of priority. And then finally you do addition and subtraction. So what does this order of operations -- let me label it -- this right here, that is the agreed upon order of operations. If we follow these order of operations we should always get to the same answer for a given statement. So what does this tell us? What is the best way to interpret this up here? Well we have no parentheses -- parentheses look like that. Those little curly things around numbers. We don't have any parentheses here. I'll do some examples that do have parentheses. We don't have any exponents here. But we do have some multiplication and division or we actually just have some multiplication. So we'll order of operations, do the multiplication and division first. So it says do the multiplication first. That's a multiplication. So it says do this operation first. It gets priority over addition or subtraction. So if we do this first we get the 3 times 5, which is 15, and then we add the 7. The addition or subtraction -- I'll do it here, addition, we just have addition. Just like that. So we do the multiplication first, get 15, then add the 7, 22. So based upon the agreed order of operations, this right here is the correct answer. The correct way to interpret this statement. Let's do another example. I think it'll make things a little bit more clear, and I'll do the example in pink. So let's say I have 7 plus 3 -- I'll put some parentheses there -- times 4 divided by 2 minus 5 times 6. So there's all sorts of crazy things here, but if you just follow the order of operations you'll simplify it in a very clean way and hopefully we'll all get the same answer. So let's just follow the order of operations. The first thing we have to do is look for parentheses. Are there parentheses here? Yes, there are. There's parentheses around the 7 plus 3. So it says let's do that first. So 7 plus 3 is 10. So this we can simplify, just looking at this order operations, to 10 times all of that. Let me copy and paste that so I don't have to keep re-writing it. So that simplifies to 10 times all of that. We did our parentheses first. Then what do we do? There are no more parentheses in this expression. Then we should do exponents. I don't see any exponents here, and if you're curious what exponents look like, an exponent would look like 7 squared. You'd see these little small numbers up in the top right. We don't have any exponents here so we don't have to worry about it. Then it says to do multiplication and division next. So where do we see multiplication? We have a multiplication, a division, a multiplication again. Now, when you have multiple operations at the same level, when our order of operations, multiplication and division are the same level, then you do left to right. So in this situation you're going to multiply by 4 and then divide by 2. You won't multiply by 4 divided by 2. Then we'll do the 5 times 6 before we do the subtraction right here. So let's figure out what this is. So we'll do this multiplication first. We could simultaneously do this multiplication because it's not going to change things. But I'll do things one step at a time. So the next step we're going to do is this 10 times 4. 10 times 4 is 40. 10 times 4 is 40, then you have 40 divided by 2 and it simplifies to that right there. Remember, multiplication and division, they're at the exact same level so we're going to do it left to right. You could also express this as multiplying by 1/2 and then it wouldn't matter the order. But for simplicity, multiplication and division go left to right. So then you have 40 divided by 2 minus 5 times 6. So, division, you just have one division here, you want to do that. You have this division and you have this multiplication, they're not together so you can actually kind of do them simultaneously. And to make it clear that you do this before you do the subtraction because multiplication and division take priority over addition and subtraction, we could put parentheses around them to say look, we're going to do that and that first before I do that subtraction, because multiplication and division have priority. So 40 divided by 2 is 20. We're going to have that minus sign, minus 5 times 6 is 30. 20 minus 30 is equal to negative 10. And that is the correct interpretation of that. So I want to make something very, very, very clear. If you have things at the same level, so if you have 1 plus 2 minus 3 plus 4 minus 1. So addition and subtraction are all the same level in order of operations, you should go left to right. So you should interpret this as 1 plus 2 is 3, so this is the same thing as 3 minus 3 plus 4 minus 1. Then you do 3 minus 3 is 0 plus 4 minus 1. Or this is the same thing as 4 minus 1, which is the same thing as 3. You just go left to right. Same thing if you have multiplication and division, they're at the same level. So if you have 4 times 2 divided by 3 times 2, you do 4 times 2 is 8 divided by 3 times 2. And you say 8 divided by 3 is, well, we got a fraction there. It would be 8/3. So this would be 8/3 times 2. And then 8/3 times to is equal to 16 over 3. That's how you interpret it. You don't do this multiplication first or divide the 2 by that and all of that. Now the one time where you can be loosey-goosey with order of operations, if you have all addition or all multiplication. So if you have 1 plus 5 plus 7 plus 3 plus 2, it does not matter what order you do it in. You can do the 2 plus 3, you can go from the right to the left, you can go from the left to the right, you could start some place in between. If it's only all addition. And the same thing is true if you have all multiplication. It's 1 times 5 times 7 times 3 times 2. It does not matter what order you're doing it. But it's only with all multiplication or all addition. If there was some division in here, if there's some subtraction in here, you're best off just going left to right.