PEMDAS is broken (and how we can fix it)

In the first week of teaching my Calculus 1 discussion section this term, I decided to give the students a Precalc Review Worksheet. Its purpose was to refresh their memories of the basics of arithmetic, algebra, and trigonometry, and see what they had remembered from high school.

Surprisingly, it was the arithmetic part that they had the most trouble with. Not things like multiplication and long division of large numbers – those things are taught well in our grade schools – but when they encountered a complicated multi-step arithmetic problem such as the first problem on the worksheet, they were stumped:

Simplify: $1+2-3\cdot 4/5+4/3\cdot 2-1$

Gradually, some of the groups began to solve the problem. But some claimed it was $-16/15$, others guessed that it was $34/15$, and yet others insisted that it was $-46/15$. Who was correct? And why were they all getting different answers despite carefully checking over their work?

The answer is that the arithmetic simplification procedure that one learns in grade school is ambiguous and sometimes incorrect. In American public schools, students are taught the acronym “PEMDAS”, which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. This is called the order of operations, which tells you which arithmetic operations to perform first by convention, so that we all agree on what the expression above should mean.

But PEMDAS doesn’t work properly in all cases. (This has already been wonderfully demonstrated in several YouTube videos such as this one, but I feel it is good to re-iterate the explanation in as many places as possible.) To illustrate the problem, consider the computation $6-2+3$. Here we’re starting with $6$, taking away $2$, and adding back $3$, so we should end up with $7$. This is what any modern calculator will tell you as well (try typing it into Google!) But if you follow PEMDAS to the letter, it tells you that addition comes before subtraction, and so we would add $2+3$ first to get $5$, and then end up with $6-5=1$.

Even worse, what happens if we try to do $6-3-2$? We should end up with $1$ since we are taking away $2$ and $3$ from $6$, and yet if we choose another order in which to do a subtraction first, say $6-(3-2)=6-1$, we get $5$. So, subtraction can’t even properly be done before itself, and the PEMDAS rule does not deal with that ambiguity.

Mathematicians have a better convention that fixes all of this. What we’re really doing when we’re subtracting is adding a negative number: $6-2+3$ is just $6+(-2)+3$. This eliminates the ambiguity; addition is commutative and associative, meaning no matter what order we choose to add several things together, the answer will always be the same. In this case, we could either do $6+(-2)=4$ and $4+3=7$ to get the answer of $7$, or we could do $(-2)+3$ first to get $1$ and then add that to $6$ to get $7$. We could even add the $6$ and the $3$ first to get $9$, and then add $-2$, and we’d once again end up with $7$. So now we always get the same answer!

There’s a similar problem with division. Is $4/3/2$ equal to $4/(3/2)=8/3$, or is it equal to $(4/3)/2=2/3$? PEMDAS doesn’t give us a definite answer here, and has the further problem of making $4/3\cdot 2$ come out to $4/(3\cdot 2)=2/3$, which again disagrees with Google Calculator. As in the case of subtraction, the fix is to turn all division problems into multiplication problems: we should think of division as multiplying by a reciprocal. So in the exercise I gave my students, we’d have $4/3\cdot 2=4\cdot \frac{1}{3}\cdot 2=\frac{8}{3}$, and all the confusion is removed.

To finish the problem, then, we would write
$$\begin{eqnarray*}
1+2-3\cdot 4/5+4/3\cdot 2-1&=&1+2+(-\frac{12}{5})+\frac{8}{3}+(-1) \\
&=&2+\frac{-36+40}{15} \\
&=&\frac{34}{15}.
\end{eqnarray*}
$$

The only thing we need to do now is come up with a new acronym. We still follow the convention that Parentheses, Exponents, Multiplication, and Addition come in that order, but we no longer have division and subtraction since we replaced them with better operators. So that would be simply PEMA. But that’s not quite as catchy, so perhaps we could add in the “reciprocal” and “negation” rules to call it PERMNA instead. If you have something even more catchy, post it in the comments below!

15 thoughts on “PEMDAS is broken (and how we can fix it)

  1. I’m struggling with this now, in fact. And it occurred to me that “parenthesis” isn’t an operation at all!

    The conventional order of operations is EMA:
    Exponents & Friends (like roots)
    Multiplication & Friends
    Addition & Friends

    And parenthesis (brackets, isolation, etc.) allow us to show when we need something else to happen – an order of the operations in a problem that is different than the conventional order.

    • Agreed, this is the perfect way of thinking about it, and I like your “& Friends” description. And you are totally right that parentheses is not an operation but a notational convention to indicate what operations to perform first.

  2. That MinutePhysics video is horrifically wrong. There are teachers posting comments saying how disgraceful the video is. It should be deleted. Now you’re adding to it. I strongly suggest you do some research, you’ll realise that you are wrong within a few minutes of starting.

    PEMDAS says (in brief)
    1) Parentheses
    2) Exponents (that includes radical)
    3) Multiplication and Division, left to right
    4) Addition and Subtraction, left to right.

    Implied and explicit multiplication are equivalent. That is to makes it simpler, It is the single biggest distinguishing feature compared with possibly every other convention.

    How can you not be aware of this?

    None of your examples are ambiguous under PEMDAS.
    e.g. a/b/c = (a/b)/c = … = ac/b because of the left to right rule.
    ab/ab = ((ab)/a)b = … = b^2

      • Thank you for pointing out what we learned in elementary school that a calculus teacher didnt understand.

        The problem with this is our teachers not teaching it correctly.

        Math is a CONVENTION everyone needs to follow it the same for it to be accurate.

        Its just sad that no one wants to teach they need to be told ……… coming from a calculus teacher!! i just cant even believe she got a job.

        • Thanks for your comment. I am questioning the acronym PEMDAS as a pedagogical tool, not the mathematical convention we use for writing unsimplified expressions. So I believe you misunderstood my post.

          Mathematical discussions are always welcome on my posts, but please refrain from using any further ad hominem attacks on either me or other commenters.

          • Lmao you didn’t even reply to the comment! He literally blew a hole in your article. You clearly state that your issue with PEMDAS is that it doesn’t account for “ambiguity” such as in the case of 6-3-2. What Chris just pointed out is that you are DEFINITELY taught to go left to right when you get to the final four steps, eliminating any ambiguity you claim there is. And you just brushed off his comment as if HE misunderstood?

          • And you are also definitely taught to switch the division of fractions into multiplication. Again, this doesn’t take away from PEMDAS, that’s a basic rule used for dividing fractions. To make matters worse, you’re a mathematics educator for Christ’s sake, spreading this confusing and incorrect crap around.

    • What you have said is actually true— that multiplication and division is in the same rank, same as the addition and subtraction. All we have to do is calculate from left to right. We are all aware that scientific calculator automatically follows the PEMDAS rule. But if the equation given is like this: 2*3÷5-6+5 , in scientific calculators, the answer would be 0.2, but what is bothering me is how come? If we use PEMDAS from left to right, it should be like this:
      (2*3)÷5-6+5
      (5÷5)-6+5
      (1-6)+5
      -5+5
      0

  3. I dont even remember PEMDAS but I excelled in Math while found that more issues in grammar. I mean I still write in broken english. More often than not on purpose but I know what the Problem is w/ PEMDAS. . . . It’s forcing broken english into a mathematical equation. As in read & write from left to right word by word rather than jumping around in a sentence. What I saying exactly?

    3+3 * 3+3 = 36
    3+ 3*3 +3 = 15
    3 + 3 * 3 + 3 = 21

    Reading & writing English is about proper spacing separating each word from one another to form a complete sentence.

    When you write
    xy3^2 + 4a + c
    I alrdy know immediately by spacing that I am adding 3 whole numbers just like Idontspeaklikethis cuz if Id ids pea klik eth is you would not understand what I am saying.

    • Utilize Spacing plus
      P – parentheses
      E – exponents
      L – left
      R – right
      Then you can manipulate the Mathematics however you choose without any confusion.

      Problem Solved

  4. This comes up a lot when CS students implement a calculator program. How about these rules instead:

    1. Apply higher-precedence operators first.
    2. Operators with equal precedence are applied left-to-right (this is where PEMDAS confuses students).
    – Exponents have highest priority: 3
    – Multiplication and division have middle priority: 2
    – Addition and subtraction have low priority: 3

    Of course, there are other wrinkles: if you have two exponents, like 2^3^4, they are applied RIGHT to LEFT, so 2^3^4 = 2^(3^4). But hopefully you won’t run into this too often.

  5. And one problem with changing conventions just because you don’t like the acronym because its “to confusing” is that’s the convention that previous math mathematicians used. So if you changed your convention all them formulas that you see that have been historically proven true are no longer accurate.

  6. Have you ever heard of PLEMDAS? That is:
    Parentheses
    Logs/Exponents (left to right)
    Mult/Div (left to right)
    Add/Subtract (left to right)

    Roots can be expressed as exponents; the log is the inverse of the exponent, as mult/div are inverses and add/subtraction are inverses.

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