The Problem with PEMDAS: Why Calculators Disagree

Some calculators say 6/2(1+2) = 1 and others say it equals 9 (similarly 8 divided by 2(2+2) can be 1 or 16 depending on the calculator). How did this disagreement on the order of operations come to be? My first PEMDAS video focused on how mathematicians, scientists and engineers interpret expressions; this video focuses on how calculators treat them. It turns out that the rule that juxtaposition comes before division is much older than "PEMDAS", and has been widely used for decades. So why did some calculator brands switch from this rule (which I call "PEJMDAS") to treating juxtaposition as the same priority level as division? And what can we do about the ambiguity?

References:

[THaWoM]

Having to go through the captions and correct all of YouTube's guesses for "PEJMDAS" has been hilarious. @14:10 "I'm firmly in favor of the pigeon toss". You can quote me on that!

[Rising_Phonix_]

I'm a programmer and it doesn't matter how simple the calculation is, I ALWAYS ALWAYS put things in paranthesis. It may be overkill, it it does 2 things. It creates a habit of using good form, and it makes the calculation crystal clear for future editors. An example for percentages is "Percent = (100 / Max) * Val". Even if PEMDAS isn't an issue, I do it anyway. It makes code easier to write, read and debug. It also creates a force of habit that makes PEMDAS bugs impossible and ensures my code operates properly every time.

[brianlara]

True Story: I lost a job offer in high school because the test checkers were using a non-scientific calculator. So I was responding to 4 + 4 / 2 as 6 while they marked me wrong because they were getting the answer of 4 ! Anyhow, I got a zero on their "test" and their calculator agreed with them ! Meanwhile, I was actually tutoring Calculus and got a 5 on the Calculus BC AP test. But yeah, there was no way to convince them that they were wrong.

[earthoid]

As an old retired electrical engineer, I have to say that PEMDAS seems so wrong that I'm surprised anyone with higher math experience would promote it.

[cmagdanz]

I'm a retired mechanical engineer, and exceptionally flabbergasted I've NEVER heard this before. I have always unconsciously used PEJMDAS (because I didn't know anything different). I simply wasn't aware of this issue, probably because I always used RPN, never entering actual equations into a calculator. I even remember when calculators weren't allowed in school ;)

[XAlphaX]

this very problem got brought up in my early highschool math classes. we were told it shouldn't matter what calculator we are using, if we aren't getting the right answer its because we weren't using enough parentheses. That habit has stuck with me and i use them like a maniac, never trusting anything's internal OOP.

[CamdenBloke]

I would definitely use parentheses in such a scenario. When I use an HP, I use RPN, so such things don't come up. When I tutored math, I always students to error on the side of more parentheses. A lot of modern calculators have formatted entry where you can have things above/below fractions, under root signs, and so on. If I'm using a calculator with that capacity, I make use of it.

[billybbob]

As a lifetime lover of math, I see this as a life or death issue worth fighting a war over. There is no excuse for calculators to disagree. It's bad enough that we cant seem to switch fully to metric. I don't care which system we use but it's pointless to have any operating order if we disagree. This is why I use parentheses EVERYWHERE when coding math for a computer. This way I can avoid implicit ordering all together. We need a calculator war and the one who loses gets their calculator destroyed / decommissioned. Kudos to Casio for throwing in the parentheses.

[dperreno]

I found a great explanation from Howard Ludwig on Quora which helps explain why NA teachers were so adamant about using PEMDAS:

"No, PEMDAS is not the truly proper order of operations. PEMDAS is an oversimplified set of rules designed to assist students (and teachers who are more education-oriented than mathematics-oriented) in advanced arithmetic and introductory algebra to keep straight the hierarchy of arithmetic operations in a compound arithmetic expression"


Note his inclusion of "education-oriented" teachers. In the U.S., many, if not most teachers below the high school level are teachers first and subject matter experts second, if at all. Meaning that many math teachers may not have degrees in mathematics, but all will have degrees in education. So they are more concerned with the process of teaching a method to use than with how things work in the real world. If they were taught PEMDAS, or their books reference PEMDAS, then that is what they think is the way it should be. Rather disappointing.

[richardward]

Thanks for the synopsis of the methods. As a Mathematician myself, I have been presented this problem several times and told each presenter to clarify what they intend. Braces, brackets, and parentheses are all free, so use them. It's is part of your job to define the problem clearly, and you cannot just assume that you are being presented the problem clearly. Always ask questions!

[johnhaswell]

When I first learned algebra as a kid in the early 80's, I was taught if there is no operator between a number and a parenthesis the 'distributive property' must be completed before clearing the parentheses. Therefore, 2(1+2) becomes (2+4) before the parentheses can be completed.

[jppagetoo]

Order of operations. A tricky subject. I have an applied math degree and I know how mathmeticians and engineers write things down. But one can easily see how it can cause errors. As a professional programmer I always use the parenthesis or break the calculation into smaller parts to make sure it is done with the order of operations I intend. I don't leave it to somebody else to determine that, that is asking for failures.

[optimist_KMA]

I am from Ukraine. as I recollect my learning and studying years, I was taught to use vertical notation of fractions as the notation of division at all times (since I got acquainted with fractions). and whenever the horizontal notation was needed, it always used as many braces and parentheses as was required to be clear what is really meant. so this video opened my eyes on a problem I have never had even a chance to experience.

[rocketsurgeon]

I learned about this while going through my education, but wasn't aware of what exactly was going on. I learned to be VERY specific with how I entered the calculations into the calculator to get the correct answer. Basically I learned "junk in=junk out". Interesting to me that the one Casio you showed does what I learned to do from experience. As another person wrote: I used parentheses almost in an overkill method to ensure that the machine understood my intentions clearly. In the case of writing these mathematical expressions, I think some of the onus is with the author of the expression to make it clear as to what the intended expression is. As with writing out languages, using punctuation is very important to the interpretation of what is trying to be conveyed. Math is a language, so the same can be said here. "Rules" are fine, but if the question is garbage, then the answer will also be garbage.

[billybob]

My HP Graphing Calculator from 10 years ago had a rather nifty feature.
There was a button to have it draw a proper 'mathematical expression' of whatever I entered, so it would draw horizontal lines, giant square root symbols, logs, and all kinds of other bits of notation.
I used it a lot to ensure the expressions I entered matched when I had written down, as it was one of those that automatically inserted explicit multiplication operators.

[privacyvalued]

When in doubt, use parenthesis. Never, ever rely on operator precedence. Extra parenthesis is only very slightly slower to parse when compiling code but will result in correct execution of that code every single time regardless of what compiles it. Once you adopt that as a core habit, you will never have issues regardless of what calculator/computing device you use. Software developers learn pretty quickly that different programming languages have different operator precedence and it is better to be a lazy parenthesis fiend than a "clever" operator precedence fiend.

[mhmt]

Thank you! I am a 57 year old American, who was taught PEMDAS growing up. However, when I went to university (for sciences), my Texas Instruments calculator routinely gave me problems. As you mentioned, I simply assumed that juxtapositions were prioritized, but did not think to check. One needed calculators for lengthy operations, especially during labs, and while checking my work—and finding discrepancies—I might have assumed the mistake was my own. Never had I considered that the trouble may lie in the calculator’s orders of operation. It was the 80’s; still the age of pencil and paper, and well before the internet.
On a lighter note: several years ago I overheard two young people pondering ‘how anyone managed to do college papers before the internet.’ Interrupting, I said, “we went to the library!”

[martinbarringer]

For calculators Reverse Polish Notation (RPN) avoids ambiguity. It does require the operator to know what calculation is required rather than relying on the calculator to interpret notation.

Once one is familiar with it, it is more efficient to use. It also has the advantage that no-one will want to borrow your calculator because they won't know how to use it!

I have been using the HP12C financial calculator for the last 45 years and it is still going strong!

[AFmedic]

What would really help would be if the teacher told their students on day 1 what system they use.
Back in the early 80's when I was going to school for electronics my teacher for DC/AC Fundamentals used PEMDAS and my teacher for Solid State Devices (Biasing transistors, etc) used PEJMDAS and we had no clue. After couple weeks we figured it and after some complaining they agreed to both use PEJMDAS and changed our grades on past assignments.

[Pajoify]

In Finland we are technically taught pemdas, but split into P E MD AS, and within MD juxtaposition takes precedence over normal multiplication and division, so technically we are taught pejmdas, just worded as pemdas.