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left to right

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I am not surprised that my removing the false information "operations with the same precedence are generally performed left to right" was reverted. So many people have been taught that false "rule" in grade school that many people insist that what they learned in grade school is true. But all mathematicians know that addition is commutative and associative and multiplication is commutative and associative, and mathematicians generally perform operations in whatever order is most convenient.

It is a bit ironic that I think 12/6*2 = 4, which is what you get when you perform operations left to right. But most physicists insist that 12/6*2 = 1. Of course, my reasoning has nothing to do with left to right. It makes sense to me that subtraction is addition of the opposite and division is multiplication by the reciprocal. It is strange that after all these centuries, there is nobody who can settle the question. Rick Norwood (talk) 10:02, 5 September 2023 (UTC)[reply]

You might see from my edit summary that my reason for the revert was that your new text was flawed, too (as was/is the previous text). If you come up with a better suggestion how to fix the false information, I won't object.
As for your 2nd paragraph above, there is no question to be settled - it is very common in mathematics that different authors introduce different ("local") conventions and use them afterwards. - Jochen Burghardt (talk) 17:14, 5 September 2023 (UTC)[reply]
Agreed. Mathematics is a human language and like any other human language there are variations and no universal "correct" standard. This article presents a set of conventions that are not universally applicable as there is not a set of rules that are universally applicable.
Also agree that the current wording is flawed. Where there is a specification to be followed (e.g. computer languages, spreadsheet and other number crunching software) almost everything evaluates addition/subtraction left-to-right (with subtraction interpreted as adding the inverse)* while other non-transitive operations such as division and exponentiation are sometimes left-to-right and sometimes right-to-left. Hence all those ambiguous memes that have everybody arguing on facebook.
In short, there is no convention for evaluating expressions like 12/6*2. And we shouldn't imply that there is.
Perhaps we should say something like addition and subtraction is usually performed left-to-right but there is no general agreement for division or exponentiation. We'd need a good source to back it up, and it may be a distraction this early in the article. Or we could just remove the sentence. Not really sure what is the best approach.
  • And when done this way, there's no need for a rule since you get the same result due to associativity.
Mr. Swordfish (talk) 18:31, 5 September 2023 (UTC)[reply]
Disagree: Each math discipline has their own rules within it. Orders Of Operations were developed to remove ambiguity when an author is unknown. It does not override a known discipline. Order Of Operations does have an agreed upon set of rules and "Left To Right" at each step IS part of those rules. 129.101.46.231 (talk) 18:19, 6 September 2025 (UTC)[reply]
"But most physicists insist that 12/6*2 = 1." is EXACTLY the problem here... The person argues that we should consider "Physics" rules when applying rules.. this is again "discipline specific"... if the problem was in a Physics book or class, then ABSOLUTELY Physics rules should apply. When the discipline is unknown, the Order Of Operations must apply and should not be altered to favor any specific disciplines. Those with the power to actually edit this page should really reconsider removing all the "special circumstances" rubbish and stick to exactly what the Orders Of Operations are. The page is misleading and catering to discipline specific "exceptions", which are not part of Orders Of Operations. 129.101.46.231 (talk) 18:25, 6 September 2025 (UTC)[reply]
Do you have a source for "But most physicists insist that 12/6*2 = 1."? 62.46.182.236 (talk) 23:00, 24 February 2024 (UTC)[reply]
Yes, very strange that this claim went unchallenged!
Unless perhaps something has changed very drastically in the decades since I studied physics that would explain such a claim? I have heard that physics is in a bad shape, and this would explain a lot. So I would be happy to be enlightened. 118.208.8.117 (talk) 04:21, 24 November 2024 (UTC)[reply]
This is a more-than-a-year-old conversation and the article has since been improved to discuss this point in much greater detail. To reiterate though, people rarely if ever write anything similar to ⁠⁠. What they do routinely write is expressions like ⁠⁠, which is interpreted to mean ⁠⁠. –jacobolus (t) 03:47, 12 December 2024 (UTC)[reply]
This whole article has gone away from actual Order Of Operations which was created to eliminate ambiguity when the author or specific discipline is unknown... instead, they have allowed "exceptions" and "special cases" to be posted that introduce ambiguity where there should be none.
The actual Order Of Operations is clear. This article should be clear... but it isn't. It is sad that we see this on such a reputable site as Wikipedia.
People really need to stop injecting their own (or other) disciplines into a neutral set of rules.
As far as the claim... it is clearly irrelevant as it talks of the rules of a specific discipline and not the rules of Order Of Operations... but I agree... Physics has changed significantly if that is what they are teaching now... but Order Of Operations hasn't changed (except on Wikipedia, I guess). 129.101.46.231 (talk) 18:34, 6 September 2025 (UTC)[reply]
Unless otherwise stated, the default convention is left-to-right. Physics journals use a different convention to save space in inline expressions. VaiaPatta (talk) 17:39, 11 December 2024 (UTC)[reply]
Exactly... "unless otherwise stated". If a specific discipline is known, the rules of that discipline applies. If there is no stated discipline or instruction specifics for a problem, then Order Of Operation is the neutral set of governing rules... and Left To Right is the default convention of OoO. 129.101.46.231 (talk) 18:37, 6 September 2025 (UTC)[reply]

129.101.46.231, you are making numerous replies to a conversation which was started 2 years ago, and in the mean time the article has been substantially rewritten (after which there were some more replies that were already somewhat out of context). The old discussion doesn't really make sense in the context of the current page, and the previous participants have probably moved on or there would be ongoing discussion in a new topic. Can you make a new topic at the bottom of the page, and state your criticisms in a way which is more matter-of-fact and less emotionally charged? I can't really understand what your concerns / goals are. –jacobolus (t) 19:12, 6 September 2025 (UTC)[reply]

Doesn't work in the real world

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Here is the problem with the order of operations: it only works in a context where it can be assumed that everybody knows it and follows it. It allows you to simplify an expression by leaving out the parentheses, which is all well and good...unfortunately, that isn’t the way the real world works.

For example: Suppose your brother-in-law agrees to fix something for you and will only charge you for the parts. He needs 3 identical parts, which cost $10 each, but are on sale at $2 off.

So he writes you a note that says: 10 - 2 x 3, meaning $8 for each part, times 3 parts, equals $24. You use the order of operations and think he means 10 minus 6 or $4. Who’s right?

Well, you could say he wrote it wrong, but that’s not correct. What he did was write it ambiguously: taken one way you get the right answer of $24, taken the another way you get the wrong answer of $4. But the bottom line is, the order of operations didn’t give you the correct answer. 2600:4040:5D3F:9A00:5D51:8DD6:75E3:9EED (talk) 14:25, 28 February 2025 (UTC)[reply]

I agree that this kind of evaluation order (simply left to right, and even with = symbols inserted to indicate intermediate results, as in 10 - 2 = 8 x 3 = 24 appears often (among non-mathematicians). Maybe we should mention this kind of habit; preferrably with a reliable source. - Jochen Burghardt (talk) 14:49, 28 February 2025 (UTC)[reply]
The "order of operations" is a loose description of the prevailing conventions in mathematics, not a prescriptive rule for how you have to communicate with your family. –jacobolus (t) 19:07, 28 February 2025 (UTC)[reply]
The Order Of Operations is not a loose description... it is exactly a set of prescriptive rules... the problem is that people have lost critical thinking. The rules are firm, but they do not override and disciplinary specific rules. They are only used on problems where the author (and therefore the discipline) is unknown. The people like the person above that try to make up a "real world" argument ultimately are trying to say that a specific discipline should be applied. The brother-in-law is know. The brother-in-law can be asked what he means... he is the "author"... SO ASK HIM. Who is right? ALWAYS THE AUTHOR... if you don't know the Author, then Order Of Operations is always right. It is really that simple, yet this article chooses to inject ambiguity by allowing various discipline specific noise into an otherwise straight forward set of rules. 129.101.46.231 (talk) 19:11, 6 September 2025 (UTC)[reply]
"any" not "and"... disciplinary rules. 129.101.46.231 (talk) 19:48, 6 September 2025 (UTC)[reply]
the brother-in-law is KNOWN (not know). I apologize for my fast typing and not proof reading before posting. I thought I would be able to edit after posting, but I guess that is only if a person is logged in. 129.101.46.231 (talk) 19:51, 6 September 2025 (UTC)[reply]
To the IP editor: you removed your comment from special:diff/1278209806, but to answer anyway: "order of opperations puzzles" on social media are at best a curious bit of internet trivia and not really worth worrying about. Shaming people about them is definitely not a good use of attention. I don't really know why these are popular, and in my opinion they're only worth mentioning here on Wikipedia to explain why not to care. As we explicitly say (quoting an expert), "one never gets a computation of this type in real life", and they are "a kind of Gotcha! parlor game designed to trap an unsuspecting person by phrasing it in terms of a set of unreasonably convoluted rules". –jacobolus (t) 04:01, 1 March 2025 (UTC)[reply]
Your brother-in-law might also drive on the wrong side of the road. (Or at least, on the other side of the road from everybody else in your country.) I don't think that means we should just abandon the road rules because people don't follow them "in the real world". Gronk Oz (talk) 02:25, 2 March 2025 (UTC)[reply]
Don’t be obtuse. The reason for math and the mathematical rules are exactly because the do work in the real world. They’re not some abstract concept. They’re rules for engineering and physics.
🤦‍♂️ Dave Dial (talk) 18:37, 2 January 2026 (UTC)[reply]
Of all the posts above, those by jacobolus best reflect my own view. I second what he said.
Rick Norwood (talk) 19:38, 2 January 2026 (UTC)[reply]

Symbols of grouping (reverted edit)

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For context: I had slightly changed the sentence "Symbols of grouping can be removed using the associative and distributive laws, also they can be removed if the expression inside the symbol of grouping is sufficiently simplified so no ambiguity results from their removal", adding "many" to the beginning. The edit reversion summary states "that e.g. the inside of a sqrt differs from brackets also doesn't need a vague pedantic caveat".

Up to that point, only three symbols of grouping had been described by the article, the first being parentheses, the second being monomials (or at least linear expressions) as argument of a function and the third being roots/radicals. Of those only the first allows simplification with distributivity and even then only in the case of a linear function applied to the parenthesized expression. More importantly, even if you ignore the less common grouping symbols including exponents and roots, fractions are a very common case and they don't allow you to simplify the denominator with distributivity either.

Finally, the last part of the sentence only applies to parentheses, for the other grouping symbols ambiguity has almost nothing to do with whether they can be removed. So I guess the sentence was supposed to mean that specifically parentheses can usually be simplified with distributivity and/or associativity and it just overshot the mark by a lot.

In any case it's a clearly false and misleading statement, even if not read as written (which would mean that not just some or most, but all grouping symbols could be removed with distributivity and associativity), and my edit was a simple and uncomplicated fix, thus unrelated to any "pedantic caveat". (Though I probably shouldn't have combined my three unrelated edits into one, that was pretty much begging for a revert.)

Since my edit wasn't accepted, I'm asking here for more ways to turn the sentence from a simple but false and misleading one into a still as simple but also true one. After all, a central point of the article itself is about how ambiguity should be avoided. In case of no feedback, I will just try another version soon. Ninjamin (talk) 03:48, 5 May 2025 (UTC)[reply]

Can you elaborate on why you think "it's a clearly false and misleading statement"? Mr. Swordfish (talk) 17:18, 6 May 2025 (UTC)[reply]
The most obvious and direct interpretation of the statement in question would be "For symbols of grouping, they can be removed with the associative and distributive laws, […]", a possibly more appropriate interpretation would be "both associativity and distributivity are able to remove [some] symbols of grouping".
The first interpretation means that at least all those symbols of grouping could be removed that are relevant in the context of the article, including the abovementioned counterexamples, which the article introduces in the directly preceding paragraph of the same section, a few lines above.
The second interpretation doesn't necessarily imply that they were able to remove all symbols of grouping, but still implies that associativity were able to remove at least some symbols of grouping, which is also not the case, because either
  1. the symbol of grouping is doing nothing else other than just grouping, which applies only to parentheses, then associativity by definition can only move the parentheses around, never remove them
  2. the symbol of grouping also applies a function to the contained expression, like a division in case of a fraction bar. In this case it can only be removed with a separate cancellation rule, never due to the change of parentheses via associativity. Examples would be canceling a function with its inverse function (in case of a double fraction or a square of a square root) or applying the knowledge that the square root of 4 is 2, which can only be applied once there are no parentheses left, not just because of moving them around.
This is why it's false. I also claimed that it's misleading and that's additionally because it implies that among the symbols of grouping it were common to find those that can be simplified with those two replacement rules, while the first rule doesn't affect any of them and the second only affects parentheses. Finally, this article is supposed to be useful for beginners and I've met many beginners that would and did simplify e.g. sqrt(x+y) falsely to sqrt(x)+sqrt(y), so it's very much relevant that it affects the square root function.
I thus suggest changing the statement either to refer specifically to parentheses or at least no longer significantly overgeneralize. Ninjamin (talk) 20:39, 6 May 2025 (UTC)[reply]
[…] and like I wrote in my post starting this section of the talk page, even distributivity only affects parentheses and the nominator of a fraction (and some more advanced symbols of grouping like integrals and some inner product notations), another way in which it's overgeneralized. Ninjamin (talk) 20:57, 6 May 2025 (UTC)[reply]
As your explanations are confusing, I went to the article, which was effectively very confusing. I would not say it was wrong, because it was too vague for that. I rewrote the sentence, and hope that the new version represents well what was intended by its author. D.Lazard (talk) 11:35, 7 May 2025 (UTC)[reply]
I think your version is both more accurate and more helpful, that should be enough. Ninjamin (talk) 14:36, 7 May 2025 (UTC)[reply]
I mainly was reverting the addition of "but for less common operations there is typically no consensus, which means that parentheses should be used to avoid confusion and misunderstanding" which is both vague and also too prescriptive for wikipedia, and doesn't really help clarify in context. But I don't think the change from "Symbols of grouping" to "Many Symbols of grouping" was to any particular benefit. However, thanks for starting a discussion. I agree with you and D.Lazard that the previous sentence was pretty mediocre. I don't think I ever paid close attention to that sentence before and am not sure where it originated, but I'll try to take another crack at it. –jacobolus (t) 15:10, 7 May 2025 (UTC)[reply]
Our articles Symbols of grouping and Bracket (mathematics) (which should probably be renamed Brackets (mathematics) since they always come as a pair in the context discussed) are both quite mediocre and should probably be merged. –jacobolus (t) 15:21, 7 May 2025 (UTC)[reply]
I assumed as much, that's what I meant when I wrote I shouldn't have combined my three edits into one. And I agree with you, the way I stated that part was too prescriptive.
My goal with that part of the edit was to prevent a beginner reading it and thinking "Oh, so I always go from left to right except for a few special operators!", when it's the other way around: The left-to-right convention is unversally accepted for sequences of additions and subtractions, but for every other infix operator I know I've found there to be a significant lack of consensus or even outright rejection. As far as I have seen, using the left-to-right rule for sequences of multiplications and divisions is mostly limited to primary education and programming languages /calculators. Even (digitally) printed mathematics using the solidus/slash for division will either define multiplication with higher precedence than division or the other way around or always use parentheses, in all three cases not following the left-to-right rule. Many common analysis teaching books for university students don't even define any infix symbol for division (no colon, no solidus) and only write it with the horizontal fraction bar or negative exponents and that's also how I've seen it in handwritten mathematics from secondary education on. If you disagree with my depiction or have any counterexamples, I would be happy to revise my view. Ninjamin (talk) 17:06, 7 May 2025 (UTC)[reply]
The first introductory section is aimed at primary education and laypeople. I put some effort (like a year ago?) into improving the section § Mixed division and multiplication, which discusses in more detail conventions found in more advanced material. –jacobolus (t) 18:22, 7 May 2025 (UTC)[reply]
Yes, I was happy when I saw that section! I just think that since already secondary education stops using that convention except with subtractions and possibly some additions inbetween, the article shouldn't imply that it was universal in mathematics, especially since school mathematics is often still taught as rigid application of rules the students don't understand. But if you think that it's unlikely someone will falsely conclude an overly general validity of the left-to-right rule, then I'll leave it as is. Ninjamin (talk) 18:36, 7 May 2025 (UTC)[reply]
The rule as stated is generally valid (for addition and multiplication of numbers), so I don't think there's too serious a problem. Anyone trying to interpret multiplication of matrices or octonions or whatever is probably not going to be trying to pedantically over-interpret the first section here. But if anyone has concrete suggestions about improvements that don't compromise basic understanding for the broadest possible audience, they can boldly make changes (which might be reverted if anyone disagrees) and we can certainly talk about it. –jacobolus (t) 21:53, 7 May 2025 (UTC)[reply]
In all of the arguments in this tread people are referring to "discipline specific" fundamentals to argue variants. Order Of Operations does not have variants. The whole idea of Order Of Operations is to be a referee when the specific discipline is not known. You are literally adding ambiguity by injecting these "some people" arguments.
WHEN THE AUTHOR IS UNKNOWN... use Order of Operations.
(NOTE: If the author of a problem is known, consult the author or refer to the discipline of the author. Do not apply Order of Operations.)
  • That order is 4 steps.
  • Each step is performed left to right before moving to the next step.
1. Groupings (list examples for claritiy) - resolve all elements inside the grouping symbols. (once complete, simplify by removing the grouping symbols and explicitly replace any operations that were otherwise implicit)
2. Exponents (list examples for clarity)
3. Multiplication (and multiplicative inverse, division)
4. Addition (and additive inverse, subtraction)
That is straight forward... everything else in the article other than examples of problem solutions using ONLY the above order of operations is a distraction and adding to ambiguity. 129.101.46.231 (talk) 19:39, 6 September 2025 (UTC)[reply]
Note: Implicit operations were never recognized in Order Of Operations. Therefore, they have no priority. Juxtaposition is recognized as multiplication, but has no defined precedence over any other multiplication or division in an unknown author problem and is worked left to right. The only ambiguity that exists is not in the Order Of Operations, but in the interjections by people that wish they were different and try to add to them. "All we need are he facts, ma'am" ~ Joe Friday, Dragnet. 129.101.46.231 (talk) 19:45, 6 September 2025 (UTC)[reply]
... THE facts, ma'am" 129.101.46.231 (talk) 19:55, 6 September 2025 (UTC)[reply]


BODMAS

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The article implies that the O in BODMAS stands for "Of" and sometimes this is given as "Order" instead - however I think this is the wrong way around. "Order" seems to be far more commonly used than "Of", a Google of BODMAS related pages shows that "Order" is overwhelmingly more common than "Of". Also one of the references used to support the "Of" version dates from 1979 so is a bit suspect for an article that should be describing current usage in my view. 123.255.61.246 (talk) 19:44, 20 May 2025 (UTC)[reply]

(Note previous discussion at Archive 3 § BODMAS' O.)
Have you done a book survey? The books and scholarly sources I found more commonly had "of", especially older ones. I'm fairly certain that "Of" was the original. It seems like some more recent authors (teachers?) who remembered the letters but not their meaning made up a new word for the O to represent, perhaps under the influence of the PEMDAS mnemonic where "E" means "Exponent", but both "O" interpretations are still quite common among the sources I looked at. I don't think a generic web search is a good way to answer this type of question. (To be honest all of these mnemonics are a harmful distraction for educators and students, and the less we think about them the better.) –jacobolus (t) 20:30, 20 May 2025 (UTC)[reply]
I'm not sure how "original" the use of Of is, I was certainly taught it was Order at school, college and university in the UK from the late 80s to the mid 90s. The first time I have ever heard it standing for Of was reading this article. That's just my personal experience of course but a quick poll of friends and colleagues from the UK found none of them remember O standing for "Of", hence my querying this statement. 2404:4400:4149:9F00:3414:127F:EF2:EEA2 (talk) 09:37, 21 May 2025 (UTC)[reply]
Can you find a reliable source clearly saying how relatively common these are / explaining the history of the variations? Otherwise, you could try to do your own survey of books and research papers. I can only tell you what I found when I went looking into this before; but I didn't do a super comprehensive survey. I'm not opposed to changing our sentence or two about this, but it should be based on something more than what one or a few people can remember about what they saw in primary school decades before. –jacobolus (t) 16:16, 21 May 2025 (UTC)[reply]
Here are some examples – "Of" is nearly universal among books I can find, especially from before the past few years, including in tertiary sources like abbreviation dictionaries and encyclopedias:
etc. Feel free to use some kind of better methodology and count if you want. –jacobolus (t) 17:10, 21 May 2025 (UTC)[reply]
The mnemonics are meant as a memory tool only to represent the steps of Orders Of Operations.
1. Groupings (Parenthesis, Brackets, etc)
2. Exponentents (Of?, doesn't make sense... or Orders, has some sense... or Indicies)
3. Multiplication (and the multiplicitive inverse, Division)
4. Addition (and the additive inverse, Subtraction)
Whether PEMDAS, BODMAS or GEMA... the letters indicate the key words. If we can use critical thinking and common sense, the word "OF" in BODMAS makes no relevant sense whatsoever... but "Orders" means something relevant... perhaps BIDMAS or BEDMAS (where "Indices" or "Exponents" are the second letter representation) would be a better representation of the flawed mnemonic?
To the point of using mnemonics at all... I strongly advise against it... especially PEMDAS and BODMAS because it adds significant ambiguity in the fact that they are 6 letters that are suppose to represent 4 steps. GEMA is far less ambiguous with it's 4 letter to 4 step, one to one ratio representation. 129.101.46.231 (talk) 20:16, 6 September 2025 (UTC)[reply]
I just linked more than 40 sources which say "O" means "Of". You should try clicking through and examining those sources. In Wikipedia, we report what reliable sources say, rather than using "critical thinking and common sense" to make up novel personal interpretations. –jacobolus (t) 20:25, 6 September 2025 (UTC)[reply]
We also report that "These mnemonics may be misleading when written this way." and "Mnemonic acronyms have been criticized for not developing a conceptual understanding of the order of operations, and not addressing student questions about its purpose or flexibility." Linking those claims to reliable sources. What individual pseudonymous or anonymous Wikipedia editors personally "advise" is not relevant to the content of Wikipedia articles. See Wikipedia:No original research and Wikipedia:Verifiability for more on this topic. –jacobolus (t) 20:28, 6 September 2025 (UTC)[reply]
As with the above, what jacobolus says above best reflect my own view and, I think the view of almost everyone who knows that they are talking about.
Great fleas have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn, have greater fleas to go on;
While these again have greater still, and greater still, and so on.-- Augustus De Morgan
Rick Norwood (talk) 19:44, 2 January 2026 (UTC)[reply]

Losing Focus

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The focus of this article is "Order Of Operations", yet several "special circumstances" are added about variant programming and other special cases. The problem here is that there are no "special cases" in Orders Of Operations. The cases being mentioned are "discipline specific" cases. Order Of Operations was developed for when the specific discipline of the problem at hand is "Unknown". It is a "referee" in a situation where the discipline specific rules are not determinable. Listing all these "exceptions" creates ambiguity in what the actual Order Of Operations are.

For instance, another person argued that "Left to Right" for Multiplication and Division or Addition and Subtraction was "false". Then they follow with an example where one discipline thinks multiplication comes first. It is NOT false... Each discipline has their own applicable rules... but in the absence of knowing what disciple we are to use to solve the problem, Order Of Operations is that governing rule... and 'left to right' is proper operation for Order Of Operations.

If this article identifies rules for specific disciplines and then link out to that discipline and those rules, great... but they do not belong in an article about Order Of Operations. The Orders of Operations were developed to ELIMINATE ambiguity and this article's authors are inserting ambiguity back in where there should be none.

PLEASE dial in what is being presented to accurately represent what Order Of Operations is and was intended to be. 129.101.46.231 (talk) 18:10, 6 September 2025 (UTC)[reply]

In common mathematical notation, the interpretation of notation is context specific and fluid and there are usually not universally accepted or explicitly written standards. Notation is based on conventions which have developed over centuries, and authors aim to have their notation be simultaneously concise and efficient for manipulation and also unambiguously understood. These two goals are often in some tension, so there are some ambiguities / inconsistencies, but usually the meaning can be inferred from context. The section § Special cases discusses some of these. We could plausibly do some re-organization, expansion, or re-writing: feel free to make concrete suggestions.
The order of operations in computer programming is different: it is exactly specified as part of a programming language's formal specification, or if unspecified there, is deterministically established by the code of the programming language implementation.
I don't really understand your broader concern. The current article is, as far as I can tell, a quite accurate reflection of current practices, and is supported by reliable sources.
The article says "Operations of the same precedence are conventionally evaluated from left to right." Are you arguing that is not accurate? –jacobolus (t) 19:24, 6 September 2025 (UTC)[reply]
You appear to be operating under the illusion that there is some official body that codifies an explicit set of rules for the Order Of Operations. There is no such organization, or if there is no one has found a reliable source that describes one and explains what those explicit rules are. If you can find such a thing, we will need to consider re-writing the article. Simply asserting that whatever rigid set of rules you might have learned in grade school is the One True Set of Rules for the Order Of Operations is not persuasive by Wikipedia standards.
Mathematics is a human language, and like any other human language there are variations, regional dialects, and opportunities for ambiguity. One can try to apply linguistic prescription to mathematical language, but that's a fool's errand. Yes, there are conventions, but there are few universal conventions. The article as currently written expresses this. Mr. Swordfish (talk) 00:39, 7 September 2025 (UTC)[reply]

Steven Strogatz and George Chrystal

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In Reference #15, Steven Strogatz's article, "The Equation That Tried to Stump the Intenet" is quoted. "But that convention is not universal. For example, the calculators built into Google and WolframAlpha use the less sophisticated convention (an order of operations understanding that does not prioritize implied multiplication over other designations of multiplication/ division) that I described in the article; they make no distinction between implicit and explicit multiplication when they are asked to evaluate simple arithmetic expressions. [...]" However, on this day of 07 Nov 2025, Wolfram Alpha's calculator actually converts an obelus division preceding parentheses to an unambiguous, vertically stacked fraction input. And unlike the conventional implied multiplication interpretation applied to lateral fractions, the stacked fraction uses the entire ratio of dividend to divisor as being the multiplier of the parentheses, which are kept at the numerator level rather than being transferred down to the denominator. It has been asserted that lateral fractions function no differently than do stacked fractions. However, this conventional implied multiplication practice does appear to demonstrate a difference between the two designations of the very same division.

George Chrystal's textbook is cited for Reference 10: "...equivalently treating division as multiplication by the reciprocal and then evaluating in any order." One common internet debate is over how to solve the division by obelus of 8 by 2, multiplying a parenthetical (2+2). The commutative property of multiplication is maintained for most combinations, agreeing with the result as given by the stacked fraction version, which functions the same as if (8/2)(2+2)=16. 8x(2+2)1/2=16. 1/2x8(2+2)=16. (2+2)x8x1/2=16. The only version of this iteration that may be interpreted by conventional implied multiplication as equaling 1 is when the fraction 1/2 precedes the parentheses... and then, only when 1/2(2+2) is designated by a a lateral fraction rather than a vertically stacked fraction. Once again, this singular iteration's IM practice is an anomaly, using only the denominator 2 of that fraction of 1/2, and relocating the parentheses to the denominator level. And so, this deviation gives the appearance of invalidating the commutative property of multiplication when implied multiplication is involved. This inconsistency only manifests when the number preceding the parentheses is a divisor, AND the division is represented in the form of a lateral fraction. The debate over whether to solve these kinds of problems via an order of operations with IM as as an integral part, versus an order of operations sans IM, would not even be an issue if the same full ratio of dividend to divisor of the stacked fraction were perceived to be the multiplier of the lateral fraction. Everyone would get the same answer of 16, anyway. ~2025-31907-19 (talk) 16:35, 7 November 2025 (UTC)[reply]

Can you be more explicit about what change you think should be made to the article? I'm having trouble understanding several of your sentences. In the first paragraph, are you saying Wolfram Alpha's behavior has changed compared to whenever this was written?
Inre your second paragraph: If we have an expression such as ⁠⁠, which according to Chrystal's explicit convention should mean ⁠⁠, it could be instead be rewritten as ⁠⁠ and then evaluated in any order — i.e. as ⁠⁠, ⁠⁠, or ⁠⁠ — because multiplication is associative and commutative. The point of the footnote is that Chrystal in practice, later in his book, also uses inline fractions along the lines of ⁠⁠ to mean ⁠⁠, without ever remarking on the difference between this practice and the initially explicitly established convention.
When we get to an internet joke/meme of something obfuscated like ⁠⁠ or whatever, there's not really a good "right answer"; the example has been intentionally constructed to poorly communicate the intent of the expression and mix together conventions from different contexts, and is much more ambiguous than anything that would ever appear in practice from a competent writer. –jacobolus (t) 17:46, 7 November 2025 (UTC)[reply]
No changes are warranted to your article.
It was negligent of me to have put the parenthetical in Strogatz's quote without clarifying that it was my own presumption that the "less sophisticated convention" Strogatz mentioned was ad order of operations sans IM prioritization. (By the way, APOST.com has an internet meme referencing a 2019 Popular Mechanics article which interviewed an officer of the American Mathematical Society about this 8/2(2+2) expression. The officer agreed that technically, the AMS position is that the order of operations should not include IM preference). But yes, it does appear that Wolfram Alpha's current translation of this expression as a stacked fraction is probably not the "less sophisticated convention" of solving problems by an order of operations not preferring IM, a convention still embraced by the other named party, Google. Implied multiplication can still be prioritized in expressions constructed with stacked fractions. Stacked fractions, like brackets and parentheses, do a much better job of clarifying numerator and denominator content of 8/2(2+2) than do such lateral divisions/ fractions lacking such brackets and parentheses. It would be my guess that Wolfram Alpha switched to stacked fraction input conversion since Steven Strogatz wrote his article, because of this improved clarity.
It was indeed understood by the footnote that George Chrystal's book declared one thing but then later illustrated solutions that contradicted that declaration. It was understood that there is not an agreed- upon convention for solving such problems. The exploration of this issue was to show that in all of the variations of the commutative property consequence and stacked and lateral fraction comparisons, the ONLY version that is subject to a different interpretation of the correct answer is a lateral fraction in which the divisor winds up on the left side of the parentheses. This typical IM interpretation contradicts the principle that lateral and stacked fractions are equivalent. This interpretation contradicts the principle that multipliers may be rearranged because of the commutative property. There may not be an established convention that declared that the multiplier is the entire lateral fraction preceding parentheses. But one probably SHOULD have been clarified and established long ago, eliminating this incompatibility with the commutative property and lateral- to- stacked fraction equivalences. ~2025-31907-19 (talk) 18:00, 8 November 2025 (UTC)[reply]

slash unclear

[edit]

typically written vertically with the numerator stacked above the denominator – which makes grouping explicit and unambiguous – but sometimes written inline using the slash or solidus symbol '/'. (§ Mixed division and multiplication)

The meaning of the phrase is unclear. Is this written by a human for humans? Voproshatel (talk) 17:13, 21 December 2025 (UTC)[reply]

Fractions are typically written stacked vertically, for example ⁠⁠, but sometimes written with the numerals "inline", i.e. aligned on the same horizontal level, for example ⁠⁠. I can try to rephrase this to avoid the word "inline", which might be the part confusing you. –jacobolus (t) 20:41, 21 December 2025 (UTC)[reply]
The cited phrase itself is clear as a statement of a possible form of notation. The problem lies in the context of the entire article, its overall meaning, and the conclusions drawn from the phrase. The cited phrase creates the impression (even though this is not explicitly stated there) that instead of a horizontal fraction one can write a slash fraction in a single line without additional parentheses, and that this will be equivalent to the horizontal fraction: . Unfortunately, the rewritten paragraph also leaves the same impression. Personally, I am too far removed from mathematics to rewrite it myself. I am only criticizing it as an ordinary reader. Voproshatel (talk) 06:16, 22 December 2025 (UTC)[reply]
The point of these two sentences is that the symbol ÷ is not widely used in mathematics after grade school. Those two sentences are not intended to be a full survey of the ways in which fraction notation can be combined. A follow-up claim that any possible example of a fraction can be written identically using either a horizontal fraction bar or a diagonal slash with numerator and denominator aligned on the baseline is not remotely implied, and anticipating that readers will infer as much seems very far fetched.
We could plausibly elaborate further about division notation and the ways it can be nested and how it interacts with parentheses in a different section of this article or in some subsection of Division (mathematics) § Notation, but in these particular sentences it is straying far off topic. –jacobolus (t) 06:48, 22 December 2025 (UTC)[reply]
Is paragraph 2 not about order of operations? Is paragraph 2 an introduction to paragraphs 3 and 4? If paragraph 2 is a transition from the symbol '÷' to the symbol '/', then maybe it would be better to put above paragraph 5 about the symbol '÷', and then paragraphs 2, 3, and 4 about the symbol '/'? Voproshatel (talk) 19:37, 22 December 2025 (UTC)[reply]
I don't understand what this comment is trying to say. –jacobolus (t) 23:23, 22 December 2025 (UTC)[reply]
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