For the benefit of the mathematically illiterate: No. “The Myth of Pemdas” is not a new Steven Spielberg fantasy film.

I’m just so tired of seeing those “suggested videos” in my Youtube feed which show supposedly confusing mathematics expressions which rely on the standard order of operations that is commonly abbreviated in public school textbooks as PEMDAS. [The original Order of Operations was a secretive Jesuit surveillance organization answering only to the Pope during the early medieval period. I think it was Tom Hanks who first discovered them.]

Anyway, you know what I’m talking about, such as with this example:

12 /2 *(2+4) = 36

But the problem is, obviously, that PEMDAS falls short with this common situation of an IMPLIED multiplication in this example:

12 / 2(2+4) = 1

And we see that kind of implied multiplication all the time in published mathematics, engineering, and computer science papers.

Yes, when you get to be my age you find all sorts of pointless gripes to complain about.

Sorry, but your second equation is false, since it violates PEMDAS. There’s no such thing as an implied multiplication; it’s just multiplication. The proper form would be 12/(2(2+4))=1.

And yet when I was in grad school in the 1970’s and 1980’s, that kind of format appeared in computer science journals on a regular basis. (Of course, the equations were slightly more complicated than the illustrations i provided above.) I would be entirely surprised if that standard has changed. Yes, it was called “implied multiplication” and from what I see online, still is.

I recall it mentioned explicitly in the submission standards for some ACM (Association of Computer Machinery) publication, probably around 1979, when I published my first paper.

Your post made me curious about today so I did a little bit of research. It appears that there has been a definite trend since my days of long ago to rely less on “implied multiplication.” I found discussions where some were claiming that implied multiplication is still common among many physicists but less so among today’s mathematicians. It is also mentioned that many favor the old convention because long equations in that format are more compact and “much easier to read”—but that the necessity of non-ambiguous use of parentheses to clarify mathematical expressions in programming languages was a powerful force for change. (Of course, in the old journals they were not so ambiguous because of how spacing was used.)

Of course, there are anecdotes often retold about equation ambiguities which caused massively expensive engineering, manufacturing, and software flubs (caused by those misunderstandings over order-of-operation issues.)

I actually remember one of my grad school professors using red-ink on my “Advanced Analysis of Algorithms II” paper where I had to apply various efficiency assessments on some B-tree reshuffling algorithm for confined memory spaces. He wrote something like, “Your discussions of the equations, especially when you show the step-wise simplifications would be far less cluttered and hard-to-read if you would follow standard implied multiplication notation. Learn from Knuth on this. He’s explicit when he needs to be but otherwise is kind to the typesetter!” He was referring to Donald Knuth’s “Art of Computer Programming” something we all owned back then. It was called the Bible of software engineering. I don’t know about today.

I always just learned it as PE(MD)(AS), with the operations within parentheses interchangeable, and the additional, obvious rule that one proceeds otherwise from start to end. Thus in 3 * 5 / 4 * 6, you do the first multiplication, then the division, then the second multiplication. Simple.

I must admit I was confused by this topic. I’d never heard of “PEMDAS”. I eventually worked out from the context that it is the equivalent of what I had been taught as “BODMAS” (“Brackets, Of, Division/Multiplication, Addition/Subtraction”), but had stopped consciously thinking about well before I finished highschool.

Looking it up on Wikipedia, PEMDAS seems to be a Franco-American thing, BODMAS is the Commonwealth equivalent.

I don’t think either are meant to be comprehensive elucidations of the order of operations, just a short and easy-to-digest Mnemonic of some of the more common and basic ones. Any video treating them as more than just that would seem to be missing the point.

I have to wonder what Youtube videos you’ve been watching to lead its algorithm to fixate like that. For myself, Youtube currently seems insistent on presenting me videos of Americans reacting to other videos about the odd, insular and/or stupid things Americans do or say (occasionally amusing, but very quickly palls). That and insisting on offering me pointless “Mixes” based upon whatever random song I last played on Youtube.

That’s because you forgot the auxiliary rule: proceed from start to end, i.e. left to right.

(((3/4)/5)/6), left to right, per the rule.

Incidentally, what does “of” mean in BODMAS? It appears to refer to exponents, but I’ve never heard that idiom. How would it be used to describe, say, 4^6?

I often find it interesting to see what A.I. engines have to say on various topics. So I asked Gemini Advanced:

Is implied multiplication using no symbol—only adjacency—still used in peer-reviewed journals when equations appear.

You’re raising a very important point about a surprisingly controversial topic in math notation!

While implied multiplication by juxtaposition (simply putting terms next to each other) is common and often convenient, it can lead to ambiguity, especially in complex expressions.

Here’s the situation in peer-reviewed journals:

No strict consensus: There isn’t a universal rule on how to interpret implied multiplication in all cases. Different journals or fields might have their own conventions.

Emphasis on clarity: The primary goal in academic writing is clarity. Authors are generally encouraged to avoid ambiguity and use explicit multiplication symbols (× or *) or parentheses to clarify the intended order of operations.

Context matters: In some contexts, implied multiplication might be well-understood within a specific field or when dealing with established formulas. However, when there’s potential for confusion, it’s best to err on the side of explicit notation.

Why it’s tricky:

Historical variations: Historically, there have been different interpretations of how to handle implied multiplication, leading to inconsistencies.

Calculators and software: Even calculators and math software don’t always agree on the order of operations in expressions with implied multiplication.

Human interpretation: Our brains tend to group terms visually, which can influence how we interpret implied multiplication.

Best practices for peer-reviewed journals:

Clarity over brevity: While implied multiplication can make equations look neater, clarity is paramount in academic writing.

Use parentheses: When in doubt, use parentheses to explicitly group terms and indicate the intended order of operations.

Define conventions: If a journal or field has specific conventions for implied multiplication, follow those guidelines.

Consistency: Be consistent with your notation throughout the paper.

In summary:

While implied multiplication is still used in some cases, peer-reviewed journals generally encourage authors to prioritize clarity and avoid ambiguity. Using explicit multiplication symbols or parentheses is often the best way to ensure that your equations are interpreted correctly.

We learned it as BEDMAS (brackets, exponents, division / multiplication, addition / subtraction).

I’ve never come across the idea of implied multiplication. We were taught the division / multiplication and addition / subtraction components of an equation proceed from left to right in absence of any brackets.

An equation like x = 3 * 5 / 4 * 6, the correct order of operations would be multiply 3 and 5, then divide by 4, and last multiply by 6. This results in an answer of 22.5.

If the intent was to do the two multiplication steps first, then it should be written as x = (3 * 5) / (4 * 6).

This coincides with how formulas work in a lot of software applications and programming.

When was still teaching I had a few grad-level international students (can’t recall which country) who described the “O” in BODMAS as “powers OF.” Obviously, ORDERS is the original intention but I suppose somewhere in their pre-college education they had learned it that way. (Perhaps the word “order” sounded inscrutable to a young person so some elementary teachers simplified it as “powers of”.)

I was definitely taught “Of” not “Order” – it appears the two may be used interchangably in the literature. In answer to John’s question: I honestly can’t remember. I haven’t thought about “BODMAS” as an explicit rule in probably something like 45 years.

I suspect that such basic rules/mnemonics will typically fade out of memory – either to be replaced by a more comprehensive and nuanced understanding of notation, if the student continues further with maths, or simply forgotten if they don’t.