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Intro to order of operations

This example shows the steps and clarifies the purpose of order of operations: to have ONE way to interpret a mathematical statement. Created by Sal Khan.

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  • blobby green style avatar for user Thoughtsofashy
    Did mathematicians choose arbitrarily the particular order of operations or there is an innate logic into it? Why did not they opt for left to right operations or exponents first then addition/subtraction, subsequently parentheses, finally multiplication/division or whatever, provided that a stable pattern was to be followed?
    (728 votes)
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  • mr pants teal style avatar for user Corey Gray
    I have always thought that within the same level of priority that the specific order (left to right, right to left, jumping around, etc.) wasn't important. At Sal says that you have to do things from left to right when you have multiple operations at the same level. At this point in the video, the problem is: 10 x 4 / 2 - 5 x 6

    Sal solves left to right: 40 / 2 - 5 x 6 = 20 - 30 = -10

    But if I don't do it in the same order I get the same answer: 10 x 2 - 5 x 6 = 20 - 30 = -10

    Thoughts?
    (167 votes)
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    • leafers sapling style avatar for user Peter Collingridge
      This confused me when Sal first said it too, but it can make a difference. For example, if the question were rearranged to:

      10 / 2 x 4 - 5 x 6

      Then you can't do 2 x 4 first i.e:

      10 / (2 x 4) - 5 x 6

      Otherwise you would get:

      10 / 8 - 5 x 6

      1.25 - 30

      -28.75

      Similarly, in the example at , you can not do:

      1 + 2 - 3 + 4 - 1 = (1 + 2) - (3 + 4) - 1 = 3 - 7 - 1 = -5
      (164 votes)
  • female robot grace style avatar for user Angela.Galileo
    The practice questions expect you to accept that a fraction bar is the equivalent of putting parentheses around the whole numerator and the whole denominator. Did Sal cover this in either of the order of ops vids? I can't find it but maybe I missed it. If not, would be a good addition to the vids.
    (78 votes)
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    • leaf blue style avatar for user Valentine
      I'm not sure that he covered this in the video, but when you have multiple operations over a fraction bar, with more operations or a single number underneath, the implication is that you are dividing the entire operation by the number underneath the fraction bar (fractions are essentially saying "the numerator divided by the denominator"). You cannot divide the operation until you have solved it, of course, so it is implied in the layout of the equation itself that you need to solve the numerator and/or denominator before dividing.
      (18 votes)
  • duskpin sapling style avatar for user Harshika M
    I have been taught BODMAS which is
    Bracket
    Of
    Division
    Multiplication
    Addition
    Subtraction .
    This is mostly the same as brackets and parentheses are the same and exponents is a different thing but then am I supposed to do multiplication first or division ??
    I have been taught that I have to divide first but here they have explained something else . What do I have to do ?
    All help appreciated😊
    (21 votes)
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    • ohnoes default style avatar for user Luis
      The way I have been taught is with PEMDAS; parenthesis, exponent, multiplication, division, addition, and subtraction. When it comes to multiplication and division, you do whichever comes first in a left to right order, same goes for addition and subtraction.
      (19 votes)
  • piceratops ultimate style avatar for user jaredona1
    i'm a bit confused... :?
    I live in england and my teacher told us to do:
    Brackets (parentheses)
    Indices (exponents)
    Division
    Multiplication
    Addition
    Subtraction
    ...so i dont do add and sub in the same group and if they are together go from left to right coz i would do the addition then the subtraction... is it different over in the US???? plz i am going mad thinking about it, which one is right????????
    (16 votes)
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  • leaf blue style avatar for user Zachary
    Could you do order of operations with fractions?
    (7 votes)
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  • duskpin seedling style avatar for user 27reillyr
    PEMDAS is what it would be for short
    (7 votes)
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  • aqualine ultimate style avatar for user 🔥Lexi🔥
    Uhm hi uhh I wanna know, uhh what is the meaning of "Exponents" It is a hard word to remember and spell. Can't they just eliminate it?
    Thanks, Sal
    - Lexi! <3
    (3 votes)
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    • blobby blue style avatar for user Skylar
      Hi Lexi! I am not Sal, but I can still help you understand exponents if you want to.
      Exponents are numbers like this: 10⁵
      The big number is the base while the smaller number floating is the number of times you multiply a base by itself. For example, in the exponent "10⁵", the expanded sentence is 10*10*10*10*10, which is 100,000. A trick only in exponents when 10 is the base is the number of 0's in the value is the small number. Like in the example, 10⁵, 5 is the small number or the exponent. So there will be 5 0's in the answer, 100,000. in exponents when 2 is the base, you just double the number the amount of times the small number, or the exponent is. For example, if the problem is 2⁵, I double 2, 4 times to get 32 (The first time doesn't count because 2¹ is just 2). also if 1 is the base, no matter what the small number is, the answer is always 1. and if the small number is 1, then the answer or value is always the base.
      Hope this helps!
      (7 votes)
  • aqualine ultimate style avatar for user Chloe :)
    How do I remember the steps, is there a trick of some sort?
    (5 votes)
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    • eggleston green style avatar for user Aiden?
      yes the way teachers taught me was PEMDAS which stands for Parenthesis Exponents Multiplication Division Addition Subtraction multiplication division left to right and same with addition and subtraction
      (4 votes)
  • leaf green style avatar for user Ashley
    at , why do you convert 8 divided by 3 into a fraction? Please dont mind my ignorance, its been quite a while since Ive been in highschool....
    (5 votes)
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Video transcript

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.