Q: How do you calculate 6/2(1+2) or 48/2(9+3)? What’s the deal with this orders of operation business?

Mathematician: Now that the Physicist and I have answered questions like these ones at least 9 times via email, I figure we should get this horrible topic out of the way once and for all with a short post.

The “order of operations” tells us the sequence in which we should carry out mathematical operations. It is merely a matter of convention (not a matter of right or wrong), giving us a standardized way to decide whether 6/2*3 is (6/2)*3 or 6/(2*3). The convention has been accepted just so that when different people see an equation they can agree on its meaning, even if they hate each other’s guts. You could come up with a different convention, but why bother? Let other people waste their time coming up with arbitrary conventions so that you can merely waste time reading posts about them.

The rules to follow are:

1. Things in parenthesis get computed before things not in parenthesis (this is because of the (little known fact) that parentheses kick ass).

2. Then, exponents get dealt with (including roots). Hence a^b*c = (a^b)*c.

3. Next, multiplication and division get carried out from left to right. So a*b/c*d is ((a*b)/c)*d. Note that division can be thought of as still carrying out a multiplication since a/b = a*(1/b), so it makes a certain sense that neither division nor multiplication takes precedence over one another.

4. Finally, additions and subtractions get carried out from left to right. Subtraction can be thought of as being like addition since a – b = a + (-b) so it makes sense that subtraction is treated on an equal footing with addition.

People sometimes use the expressions PEMDAS, “Please Excuse My Dear Aunt Sally” or “Puke Exhaust Mud Dolls Autumn Sasquatch” to remember these rules. These all stand for “Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction”. It is easy to get confused by these mnemonics though because they don’t capture the fact that multiplication is treated the same way as division (carried out left to right), and addition is treated the same way as subtraction (carried out left to right). That’s why I prefer: “Puke Exhaust Mud And Dolls Autumn And Sasquatch”

One interesting thing to note about the order of operations is that there is nothing interesting about them whatsoever. But that implies there is something interesting about them. This is a contradiction, and hence the order of operations form a paradox. They are also about as fun as getting to watch a fish crawl up the stairs as a reward for answering math questions.

But, that won’t stop me from giving an example. Consider:

a^b*c/d+e-f.

Without a convention, it has many possible interpretations, such as

a^(b*(c/(d+e)))-f

or

(a^b)*((c/(d+e))-f)

which could be horribly confusing for people trying to communicate with each other. But, the correct interpretation according to the rules mentioned above, is:

(((((a^b)*c)/d)+e)-f).

When in doubt about what the orders of operation are for an expression, there is a simple solution which works even better than tattooing PEMDAS to your face. Just paste the expression into the google search bar and hit “search”. For instance, if I paste in:

4^2*3/6+1-5

it gives back

(((4^2) * 3) / 6) + 1 – 5 = 4. 

Not only is this the correct answer, but it shows you explicitely the order of operations that were used, so it’s useful for learning.

Google even gets this one right:

4^2*3/6 + 1 – 5/14*3 + 6/10 – 4*2/14*6^3/18 + 14

which is the kind of question little sisters give you when they are old enough to finally sort of get what a mathematician is.

It’s important to note that not all calculators handle the order of operations in the standard way. Why? Because people who make calculators hate children. The preceding sentence was not intended to be a factual statement.

Also, a lot of programming languages use the standard order of operation rules, but then you have exceptions like Smalltalk.

Oh, and in case you were wondering:

a^b^c  = a^{b^c}.

This entry was posted in -- By the Mathematician, Conventions, Math. Bookmark the permalink.

172 Responses to Q: How do you calculate 6/2(1+2) or 48/2(9+3)? What’s the deal with this orders of operation business?

  1. mathman says:

    Listen closely: 2x = (2x) = (2 * x)
    If you think that 2x does equal (2 * x), then I have news for you.
    Also, as I have stated: You cannot change the coefficient of (2+1), which is clearly 2, into (1/2).
    Just to re-iterate:
    2x = (2x)
    So if you would like to substitute something different into the original equation (or expression), then you require parentheses.
    Like I stated in me previous post:
    6 ÷ 6 = 1
    Let 6 = 2 * 3
    Now I have: 2 * 3 ÷ 2 * 3 = ?
    This is basic algebra and mathematics techniques. You are not allowed to substitute “willy-nilly”. :)

  2. RB says:

    @mathman

    Just a quickie here for you. 6÷6=1 is absolutely not the same as 2*3÷2*3 unless you use parentheses. * and ÷ have the same precedence in Order of Operations. 2*3÷2*3=9. Don’t be “willy nilly”.

  3. mathman says:

    @RB: My point exactly. You can’t just insert a multiplication sign between a set of factors, coefficients, and literals without containing it parentheses.
    (6y²z⁵ ÷ 2xyz²)² is not the same as (6y²z⁵ ÷ 2 * xyz²)²

  4. MK says:

    @mathman: I think the problem comes from IMPLIED meaning, and since people don’t want to write parenthesis around everything they do, they just assume them and move on. When you write (6y²z⁵ ÷ 2xyz²)² what is usually meant is (((6(y²))(z⁵)) ÷ ((2x)(y)(z²)))², which is a lot of writing for what should be something pretty straightforward. But, written out inline without those parenthesis, it SHOULD be interpreted as ((6)*(y²)*(z⁵) ÷ (2)*(x)*(y)*(z²))². But, because of a convention that teaches us to imagine the parethesis instead of writing them, it creates confusion at the elementary levels with problems like 6/2(2+1). You either follow the order of operations to the letter, from left to right without implied parenthesis: (((6) / (2)) * (2+1)) OR ((6) * (½) * (2+1)), or you treat is as an algebraic coefficient, which would really read as (6 / (2 * (2+1))).

    Imagined parenthesis also leads to bad math reasoning like:
    6 / 6 = 1
    6 / 3 * 2 = 1
    now, following order of operations
    (6 / 3) * 2 = 1
    2 * 2 = 1
    4 = 1 ??
    look, i just broke math!

    Moral of the story, parenthesis are probably the most overlooked but important tool you can use to say what you really mean. The reason there is so much confusion is, in my humble opinion, the somewhat necessary teaching (so many parenthesis!) of a convention that tells you to assume.

  5. mathman says:

    Thank you for commenting, but it actually should ~not~ be interpreted as you wrote it. An expression as written, should never be presented to an elementary student in the first place. Also, mathematicians, physicists, chemists, etc, do want to write out dozens of parentheses, as you can imagine, because it would like your example. We string together variables and understand that it is one unit to avoid such things. Putting the multiplication symbol between them all accomplishes nothing but confusion, and would require parentheses around the entire group, which makes it look ugly and slows down the ability to read the information quickly. A product of coefficients is the coefficient of the remainder of them, and so on. They are also commutative, as in abc = bca = cab … It isn’t a terminally difficult concept. 2 = (2), right? There are always implied parentheses around numbers. The same holds true for groups of variables and the like. I am not sure if this got omitted in schools in the last 20 years, but it is a very basic algebraic concept.
    Ciao

  6. MK says:

    I was speaking in a purely elementary fasion for interpreting the problem. If you are just holding order of operations as the highest authority then that is how it ~should~ be read, but the algebraic convention says otherwise because we assume those parenthesis!
    As for the multiplication; it is commutative, but the division has to be handled with much more care. Without those parenthesis, are you multiplying (since division is a form of multiplication and is, in my opinion, a much simpler way to avoid confusion) by 1/2, or is it multiplying by 1/(2xyz²)? in algebra, this is a very easy answer, because we teach the CONVENTION of using coefficients and implying parenthesis (again, for good reason).
    But, when you take that same logic and apply it to elementary math, you end up with a disconnect that crops up with cases like 6/2(2+1). without more clarification, it is pretty ambiguous whether the WRITER of the equation meant multiply by 1/2 or 1/(2*(2+1)). If you convert it to an algebraic form, then it is (a/(b(b+c))), which is 1. But if the person is writing from/for an elementry level, the question would instead be written out as (a(1/b)(b+c)), which is 9. Without knowing the context of the question, and with two different conventions (implied vs absolute parenthesis) to chose from, how can you be sure what the author meant?
    The solution is either rewrite the original question to include parenthesis and remove any ambiguity, standardize the convention and teach elementary math using the same assumptive parenthesis as algebra, or teach numerators/denominators instead of the division symbol at the elementary level.

  7. MK says:

    or, if we boil the problem down to its simplest form, we have to ask whether the “/” sign means “÷” or “everything following is the denominator”, since unfortunately it seems to be used interchangebly by people to mean either one.

  8. Mitchell says:

    @MK if youre going to start factoring 6 then you need to do it correctly.

    6/6=1
    6/2*3=1 you went wrong right there
    6/(2*3)=1

    I believe I found a better way to understand this
    for people who always get 9

    Lets write this out with variables.

    6/2(2x+x)=y
    simplify
    6/2(3x)=y

    is the next step 6 * 3x = 2y or is it 6=2(3x)y or is it 6/3x=2y or is it 6/3x=y/2

    the answer is 6=2(3x)y or 6=6xy then divide both sides by 6 and you get xy=1 already knowing that x=1 therefore y=1 and the math checks itself.

    do this expecting to get 9 for y and you will not end up with x equalling 1

  9. Ray says:

    This whole equation was originally written as 6÷2(1+2) rather than 6/2(1+2), so somewhere in the hundreds of posts, someone could not figure out how to write ÷, so it was transformed. Having said that, order of operations says that we first take care of the (1+2) which gives us a next step of 6÷2(3). Since division is considered multiplication and the rules for Order of Operations indicate that both multiplication and division have the same precedent and should be followed from left to right, the division would be handled next giving an intermediate step of 3(3). The last operation to occur would be the multiplication, giving us an answer of 9.

  10. mathman says:

    Again. not a dilemma of order of operations, but rather one of operators and operands. 2(3) is not simply 2 * 3.
    When you use that rule, it states that, for example, 2a = the product of 2 * a. That is what everyone is missing. 2(3) is the same as 2a where a = 3, except that we can’t write it the same way, that is, without the parentheses since it is 2 single digits adjacent to each other.
    REMEMBER THIS: 2(3) = (2)3 = (2 * 3)
    just as abc = bac = cba, etc.
    Anyone who has spent any amount of time in some serious education, whether it be calculus, stats, chemistry, physics, or mechanics, would know this. That is the frustrating part of it all: Those who haven’t had the opportunity to be immersed in such depths of any of the degrees listed, yet they consider themselves experts.
    I repeat: 2a = the PRODUCT of 2 * a, which means it is (2 * a).
    Someone, somewhere, said it was ok to substitute back 2 * a into the original expression, when it is not. But hey, they read it on the internet right? So it must have been correct.
    It is not 9, folks.

  11. Mitchell says:

    @Ray Did you get that ^^^^ 2(2+1) is the same as [2*(2+1)] and that is why you are wrong. You do not separate the individual numbers of the expression. That’s a lame excuse to say that the problem is that people confuse the division symbol for a fraction line….a fraction line is division so smack your head for me please. This is why there is something called implied multiplication implicated by juxtaposition. The (2+1) belong to the 2 because the math implied has been simplified removing all extra operands. I’ve been telling my teachers they were wrong about this for years. And the problem is that people dont even care to check the math. Try writing this equation using 1=x and trying to solve for y=9. 6/2(2x+x)=y Solve for x and if you dont get 1 when using 9 for y then your are wrong, and since I’ve already done this I’ll just let you try it out on your own.

  12. MK says:

    @mitchell: i understand that, which is what i was trying to show. The incorrect use (and sometimes assumptions) of parenthesis lead to math errors such as the one i explained, many of which showed up in the comments on this thread.

    @mathman: that is essentially my point. with an elementary problem like that and varying math levels, it is difficult to know if the original author meant 6÷2*3 or 6÷(2*(3)). I don’t think either answer is wrong as much as the equation needs to be clarified a little to remove possible ambiguity. I agree that the question, as written, should be interpreted as 6÷(2*(3)), but that doesn’t preclude the possibility that the author intended 2(3) to be shorthand for 2*3 (as it often is in lower level math). This doesn’t make either convention necessarily wrong, it just means you have to know which convention to apply (though i do vote for chosing only one to fix situations like this).

    Or, to put it a slightly different way, if i write the date 07-10-13 in america, most people will read it as July 10th, 2013. But for someone from europe who is visiting, they might interpret it as 7th October, 2013. Neither is wrong, you just need to know which was intended.

  13. mathman says:

    This intention of the author is neither here nor there. :) The problem is just that: a problem. I have had calculus profs that “intended” a solution, but when I solved otherwise, and was marked wrong, I showed the error and was awarded the marks. For the problem at hand, mathematical techniques must be ignored for the solution to be something other than 1 or 2 (for the corresponding problem).
    However, this ‘problem’ should never be presented to an audience that isn’t knowledgeable enough to solve it properly, which happens to be about 2/3’s of the internet by reading most of the polls online. Let me correct myself with (2/3)’s … ;)
    Just because most people get it wrong or don’t know any better, doesn’t make this a convention issue, but rather a educational one.

  14. Mitchell says:

    I have never had a problem discerning that when a number is placed immediately outside of parentheses it and the values inside the parentheses are one single expression that must be simplified before the equation can continue its order of operations. As stated above and quite clearly there is no need for clarification on the syntax of this equation, the clarification needs to be taught in school that 2(2+1) is shorthand for [2*(2+1)], it is not all that hard.
    6/2(2+1)=1
    6/2*(2+1)=9
    its that simple

    basically you are trying to allow people to be correct in assuming that the equation is equivalent to 6/2*(3) instead of 6/2(3). There is a profound difference and it needs to be taught. Science cannot afford mistakes like this. I hate misinformation and allowing people to believe the answer can be 9 or 1 is WRONG. Theres is a perfectly simple explanation for why but people are willfully ignorant because they were taught the ORDER OF OPERATIONS in school so it must be absolutely true. It is true but the fact is that no matter what level of math you have learned all RULES apply at ALL times. I’m and getting tired of people cherry picking the rules so they can make 9 fit without even rearranging the equation to see if 9 actually works which it doesn’t.

    Lets have you work backwards. We’ll start with:
    6/2(2x+x)=y solve for X and then insert 9 for y and tell me if you get 1 for x
    6/[(4x+2x)]=y
    6/6x=y
    6=6xy
    1=xy
    y=x

    so how can y be 9 if we know x is 1

  15. Edward says:

    @Mitchell: I agree with you that the answer is one. However, the original article we are replying to indicates the answer should be 9, and he is a professional mathematician. So I guess it isn’t cut and dry like we wish. I believe the 2 is a coefficient of the parenthetical expression, and thus is linked to it in priority over PEMDAS. But calculators and sites as he describes in the article don’t agree.

  16. Mitchell says:

    Calculators are programmed. They are not absolute on correct mathematical procedure. Computers can only do what we tell them to. If the calculator operand precedence is flawed within the programming how could we possibly expect people to know the right answers to questions like this? I do agree that the problem here is the education of the math involved. People think that they only have to apply the rules of arithmetic PEMDAS when it requires higher level math. The people who get 1 have A)Higher IQs
    B)Better Education
    C)Common Sense that dictates 2*(1+2) is completely different from 2(1+2)

  17. Raed Khoury says:

    6÷2(1+2) How do you read this? is it (6÷2)(1+2)=(3)(3)=9 which means you have created 2 new parenthesis which means you have changed the equation and decided to include the 6 in the multiplication with the (1+2) which means you have changed the answer and if we were dealing with millions of $$ you would .. well you get the picture I hope. Or do you read it as 6÷[2(1+2)]=6÷[2+4]=6÷[6]=1 and in this case we also added some parenthesis so we could come with the OTHER answer. I have no problem going left to right as long as the problem have been simplified and completely. so lets do this again without changing or adding anything to this simple problem : 6÷2(1+2)=x using simple Algebra I learned in 6th grade: 6÷ (2 times 1+ 2 times 2)=x which means we took the 2(1+2) and simplified it to its least confusing shape. now we have the following: 6÷[(2×1)+(2×2)]=x are you still with me? now we have 6÷[2+4]=x Plz don’t fall asleep on me am almost done: x=6÷6= 1
    Left to right, up and down, dogie style, what ever pleases you simplify first

  18. RB says:

    @Raed: Is 2(1+2) the same as 2 x (1+2) (x as in multiply operation)? 2(1+2)=6 and 2 x (1+2)=6 Since they both are 6, we should agree that it is the same. Now try this using Order of Operations: My tablet will not allow me to use ASCII, so I will spell out obelisk. 6 obelisk 2 x (1+2) OofO states division and multiplication have same precedence, so this is 6 divided by 2 times (1 plus 2) and by OofO, next step is 6 divided by 2 times 3. Next step 3 times 3 and final answer is 9.

  19. RB says:

    Back to my laptop. Just to reiterate that last problem done without the assistance of ASCII characters. We will use basic Pre-Algebra for this one. As stated before, we know that 2(1+2) = 6. We also know that, because of the implied multiplication of the (), that this is the same as 2 x (1+2) = 6. Now let us write this with all out in the open, including the multiplication between the 2 and the (1+2). 6 ÷ 2 x (1+2) is still the same problem, is it not? Now let’s read up in our Pre-Algebra book and it says that we first clear the parentheses. This leaves us with 6 ÷ 2 x 3. Now we again look at the Orders of Operation, same book, and it tells us that both division and multiplication have the same precedence. After all, it is the same as 6 x 1/2 x 3. But I digress and going even this far may confuse some people. So, having Orders of Operation in mind, and looking at 6 ÷ 2 x 3, we know that we must now go left to right because of equal precedence and we now have 3 x 3. Last step, 3 x 3 = 9. Or, is it just better to leave this as too ambiguous to please everyone.

  20. Mark says:

    Excel gets 9. When I was in school for Engineering x(2)=x*2. Excel does the same thing… So does EES and Mathcad… And my Ti92 plus!

  21. mathman says:

    Since when do we simplify parentheses by inserting a multiplication symbol between the factor and parenthetical? Have you ever seen an example or proof of this sort of thing in a published text? I sure haven’t.
    The proper way to simplify parentheses is:
    a(b+c) = (ab + ac)
    We are allowed to replace this entire group of terms, according to the definition of math properties.
    6 ÷ a(b+c)
    6 ÷ 2(2+1)
    or, if a is an expression of more than one term or operators, it is always contained in parentheses itself (a + b)(c+ d) or other variations.

  22. Since
    6÷2(1+2) ≠ (6÷2)(1+2)
    Because it is NOT implied
    Do NOT imply a parenthesis which is NOT there
    Since it is NOT shown
    AND
    Since
    (6÷2)(1+2) = 9
    Therefore
    6÷2(1+2) ≠ 9
    Since
    6÷2(1+2) ≠ (6÷2)(1+2)
    THAN
    6÷2(1+2) = 6÷(2+4) = 6÷6 = 1
    Where the ‘2’ is part of the parenthesis
    The problem is that we are implying that
    6÷2(1+2) = (6÷2)(1+2)
    However
    The equation is NOT written that way!
    There is NO parenthesis sign () there!

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