A world without order (of operations)

What would such a world look like? Like this:
(((3\times(x\times x)) – (7\times x)) + 2).
$$What a world it would be! A world without ambiguity! A world where PEMDAS would just be P! A world where they would have to relocate the parenthesis keys to a more convenient location on the keyboard!

Parentheses, and order of operations, tell us how to read the meaning of an expression, how to parse it, not what to do with it. In the expression above, every matched pair of parentheses contains something of the form $$
(\mbox{blob}) * (\mbox{another blob}), \qquad (\mbox{where * stands for $+$, $-$, or $\times$}),
$$ unless the blobs are just numbers or letters, in which case we don’t surround them with parentheses. We always know exactly what things we are adding, subtracting, or multiplying. Starting with the outermost parentheses, we see it contains the sum of 2 and a blob. Looking inside that blob we see that it contains a blob minus another blob. And so on. The structure of the expression can be represented in a diagram:

So what is order of operations about, and why do we need it? Well, that’s a lot of parentheses up there, so it is useful to have some conventions about when things are understood to be a blob, without actually putting in the grouping symbols (blobbing symbols?). First, any sequence of multiplications and divisions is understood to be a blob (that’s the precedence of multiplication and division over addition and subtraction). Second, in a sequence of additions and subtractions, or of multiplications and divisions, you read from left to right. (Actually, there is disagreement about this last one in the case of multiplication and division, but never mind.) The first rule allows us to write the expression above as
((3\times x\times x- 7\times x) + 2).
$$The second rule allows us to leave out all the remaining parentheses. And, of course, we have other conventions about representing multiplication by juxtaposition, and about exponent notation, which allow us to write
3x^2 – 7x + 2.

Calling it order of operations is problematic because it can be misconstrued as suggesting that there is a specific order in which you must perform operations. There isn’t, except insofar as you sometimes have to wait to perform an operation until you have calculated all the blobs in it. But, for example, there is no law that says you have to do the multiplications first in $101\times56-99\times56$ and, in fact, it is more efficient to factor out the $56$ and do a subtraction first. Order of operations tells us how to read this expression: it’s a difference of two products, not a product of three factors the middle one of which is a subtraction. But it doesn’t tell us how to compute it. The word “order” in “order of operations” is best understood as referring to order in the sense of hierarchy, as in the diagram above.

Outside of textbook school mathematics the order of operations is a matter of common law, not constitutional law, and it’s a bad idea to make a federal case out of it on assessments. For example, dinging a student for interpreting $x/2y$ as $x/(2y)$ rather than $(x/2)y$ would be unreasonable; many scientists would do the same thing. If there is any danger of ambiguity we should put the clarifying parentheses in.


A few final thoughts:

    • thanks to Brian Bickley for suggesting the topic for this post
    • there’s a nice discussion of the history of order of operations over at the Math Forum
    • and bonus question: do we have to give multiplication precedence over addition? Could we do it the other way around?

8 thoughts on “A world without order (of operations)

  1. It can be demoralizing to leaf through a middle-school algebra textbook and see the amount of stress being laid on the order of operations, as compared to the amount of stress that’s being laid on the *properties* of operations.

  2. Could you describe this scenario? Which plays out first in order of operations?

    6divided by 2 times quantity 2 plus 1 close quantity.

    Once you perform the inner operation, does distribution precede the division – which is my question, does distribution require its own order of operations?

    • This is a good example of an ambiguous expression. The only way to be completely clear about what you mean is to put in enough parentheses that conventions about order of operations are not needed. Some people would think this means (6/2)*(2+1). Others might think it means 6/(2*(2+1)). It’s up to the writer of the expression to make that clear, not the reader. Once the writer has made it clear, then the order in which things are done is a choice of the reader. Going with the second interpretation, they could add the 2 and the 1 in the parentheses first, then multiply by the 2, then divide the result into the 6. Or they could distribute the 2 and get 6/(4 +2), then add the 4 and the 2, then divide the result into the 6. Order of operations does not tell you which “plays out first.” It just tells you what the expression means.

      So no, there is no need to make a rule about whether you distribute first then add, or add what’s in the parentheses first and then multiply, and in fact such a rule would be harmful because there would be some occasions when you wanted to do one and some when you wanted to do the other. Which you choose depends on your purpose.

      I wouldn’t talk about distribution as an operation at all, but would rather talk about the distributive property, which tells me that $a(b+c) = ab + ac$. What I do with that information is up to me.

      • Parentheses remove ambiguity, and just using that “rule” instead of the chants for order of operations saves a great deal of class time and removes some of the clutter in the curriculum.

        Related:  We’re working on a computer science curriculum that uses a programming language that is unambiguous about what gets done when and doesn’t use parentheses (well, you may say  that it uses them in disguise).

        The course is called \emph{Beauty and Joy of Computing}:


        The language (like many programming languages) \emph{requires} you to say what you mean.  As an example, here are the two versions of the calculation posted by Bob Knittle.


        And you can run the calculations to see if they give you what you want.


  3. There’s been some discussion of this over on the Illustrative Mathematics Facebook page, which prompts a few further thoughts. There are two ways of eliminating ambiguity in an expression: one is to use order of operations (even in situations like $6/2y$), and the other is to use parentheses. Of the two, only the second is universally accepted and understood. Once you get beyond using the symbols $\times$ and $\div$, and start using juxtaposition to indicate multiplication and a slash to indicate division, you can’t expect agreement. Most scientists, engineers, and mathematicians would interpret the expression $3a/2b$ as $(3a)/(2b)$. Ideally they would always write the latter, but they will probably forget sometimes, and if you take a strict left-to-right approach and rewrite it as $3ab/2$ on a calculus exam you are going to lose points. And your professor might not be swayed by “that’s the way I learned it in middle school.”

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