Talk:Order of operations/Archive 4

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What a mathematical operation is

In mathematics, an operation is a kind of function. If the function has one input, it is called a unary operation. If it has two inputs, it is called a binary operation. Taking the opposite of a number is a unary operation. Taking the sum of two numbers is a binary operation.

Parentheses are symbols of grouping. They are not a function. Their use does not depend on the expression inside them.

This is so badly taught in American schools that I have to work hard to train students to use parentheses correctly. But many people resist, and continue to insist, to give just one example, that problems must be worked from left to right. They also insist that you must "do" what is inside parentheses first, as if there were a way to "do" x + (yz). The correct statement is that expressions inside symbols of grouping can be considered as a single mathematical object. The order of operations tells us that x + (yz) = x + yz so the product yz is considered as a single term in the sum. Rick Norwood (talk) 10:39, 21 April 2023 (UTC)

Your first two paragraphs are correct. I don't think anyone is arguing against that here.

This is wikipedia. Our job as editors is to reiterate what is presented in the source material. It is not our job to go on some kind of crusade to "fix" the American education system or to engage in hair-splitting semantics over language.

See https://mathworld.wolfram.com/Precedence.html and https://mathworld.wolfram.com/Parenthesis.html, in particular "Parentheses are used in mathematical expressions to denote modifications to normal order of operations." While parenthesis are not themselves operations, they are used to modify the order of those operations. Hence, every treatment of order of operations discusses parenthesis.

All that said, the purpose of this talk page is to discuss improvements to the article, not general discussion of the topic itself. Mr. Swordfish (talk) 13:33, 21 April 2023 (UTC)

I have no problem at all with the use of the quote you gave, either before or after the list of the order in which operations are carried out. In face, I'll put in it myself. My objection is and always has been calling parentheses an "operation". It is not "hair-splitting" it is just good mathematics. Rick Norwood (talk) 22:11, 21 April 2023 (UTC)

I see that you have again removed parentheses from the definition. There is NO CONSENSUS here to do so. Please stop your disruptive editing. Mr. Swordfish (talk) 22:24, 21 April 2023 (UTC)

You are the only person who insists on calling parentheses an operation. The change I made was a change you yourself suggested, so it is hard for me to see how I am being disruptive. Rick Norwood (talk) 22:26, 21 April 2023 (UTC)

You have repeatedly falsely claimed that I "insist calling parentheses an operation". I do not claim that parentheses are an operation. Neither has anyone else here. Nor does the article claim that parentheses is an "operation". Please stop falsely accusing me of things.

To recap: a few days ago you edited the article to remove parentheses from the definition. It was quickly reverted (not by me). In the ensuing talk page discussion, every editor (D.Lazard, Dhrm77) disagreed with your edit, not just me. Now, please respect consensus and recognize that you are alone in your opinion and leave the article in it's current state until we reach consensus to change it. Mr. Swordfish (talk) 23:01, 21 April 2023 (UTC)

I hope someone here knows how to resolve this situation. Mr. Swordfish's description above is totally incorrect, but he will no doubt claim that I am the one who is incorrect. The facts are clear, as anyone reading the material above can see, and I do not want to continue this revert war. I hope someone knows how to end it without leaving the mathematically wrong version Mr. Swordfish insists on in place. Please note that the reversion he is now reverting using his own quotation from Mathworld, and still he will not accept it. Rick Norwood (talk) 23:09, 21 April 2023 (UTC)

The way to resolve it is to reach consensus on the talk page i.e. convince other editors that your revision is an improvement. The way not to resolve it is to continue re-inserting your revision without reaching consensus.

I've opened up an ANI at https://wiki.riteme.site/wiki/Wikipedia:Administrators%27_noticeboard/Incidents. Perhaps they can help. Mr. Swordfish (talk) 00:09, 22 April 2023 (UTC)

The sentence just beneath the ordered list in the Definition subheading does kinda read like parentheses are being called operators: This means that if, in a mathematical expression, a subexpression appears between two operators, the operator that is higher in the above list should be applied first. I think keeping parentheses on the list best serves the reader, although I don't feel strongly about it, and perhaps the quoted sentence could be reworded to clarify that not every entry on the ordered list of operations should be construed as an operator. Folly Mox (talk) 02:32, 22 April 2023 (UTC)

I agree that it is possible to read the current wording as implying that parentheses are operators. I don't read it that way, but can see that some may. I tried to address that in my most recent edit, but that was reverted. I would support more careful wording to avoid creating the impression that parentheses are operators.

Including parentheses as item #1 is absolutely standard as found in all the reliable sources cited; removing it would require finding more reliable sources that contradict the ones we have. I doubt that's possible, but I'll look at any that are presented. Mr. Swordfish (talk) 02:47, 22 April 2023 (UTC)

Personally, I feel replacing the second instance of operator in the quoted sentence with operation would be sufficient, but I'm not a specialist in this topic area. I might also consider internationalising the note about parentheses, since while PEMDAS is indeed the American mnemonic, it is not the mnemonic without qualification, as the lower bits of the article bear out. Reaffirming that usability indicates inclusion of the brackets / parentheses. Hope consensus can be reached here without any more exasperation. Folly Mox (talk) 03:54, 22 April 2023 (UTC)

There is clearly a consensus that edit warring is counterproductive. And yet, the most recent edit, by someone who signs themselves 2601:18f:107f:e2a0:7142:367:472:ca68, who has never before edited anything on Wikipedia, has simply restores Mr. Swordfish's version of the article. Rick Norwood (talk) 09:42, 22 April 2023 (UTC)

While almost all elementary schools teach PEDMAS, here is an example of the problems that causes.

https://slate.com/technology/2013/03/facebook-math-problem-why-pemdas-doesnt-always-give-a-clear-answer.html

Rick Norwood (talk) 09:56, 22 April 2023 (UTC)

D. Lazard's recent edit

I find D. Lazard's recent edit entirely acceptable and hope this ends the edit war. Rick Norwood (talk) 12:14, 22 April 2023 (UTC)

I also agree with that edit. Mr. Swordfish (talk) 12:54, 22 April 2023 (UTC)

I would consider internationalising the article by using "brackets" rather than "parentheses" (obviously keeping the existing note as to what PEDMAS stands for), as parenthesis is really only used in the USA in the English-speaking world. Indeed, parenthesis actually redirects to Bracket. Certainly, the sentence "the parentheses may be replaced by brackets or braces to avoid confusion", with "brackets" redirecting to "square brackets", is particularly confusing. Black Kite (talk) 19:14, 22 April 2023 (UTC)

PEDMAS and all the other acronyms going all the way back to My Dear Aunt Sally have clearly done more harm than good, but will probably exist longer than the human race exists. I'll only point out that half the people who were taught PEDMAS or BEDMAS think a/bc = a ÷ b ÷ c and the other half think a/bc = a ÷ b ∗ c and both sides, having been misled in grade school, are absolutely certain that they and only they are correct. Rick Norwood (talk) 19:20, 22 April 2023 (UTC)

I just discovered that Wikipedia math will not allow either. This is probably a good thing. Rick Norwood (talk) 19:26, 22 April 2023 (UTC)

The Manual of Style says:

When an English variety's consistent usage has been established in an article, maintain it in the absence of consensus to the contrary. With few exceptions (e.g., when a topic has strong national ties or the change reduces ambiguity), there is no valid reason for changing from one acceptable option to another.

This article uses US English (and has from the beginning), so the MOS tells us to continue using it. Hence "parentheses" instead of "brackets" in most uses. That said, we should modify the intro section to indicate that "parentheses" (US English) is the same thing as "round brackets" (UK English). Simply referring to "square brackets" in the final sentence should clear up any confusion. Mr. Swordfish (talk) 21:36, 22 April 2023 (UTC)

Is there a rule in mathematics that arithmetic must be done from left to right except where the order of operations says otherwise

There is no such rule in any reliable source. That rule appears in countless grade school text books, but does not appear in any book written by a reliable professional mathematician. Sadly, grade school level math books are full of false statements. Books written for children are not reliable sources.

My children were taught in grade school from a textbook that said 5 - 1 + 1 = 3 because "Aunt" comes before "Sally". The textbook was written by a leading math educator, but it was wrong.

No serious mathematician would ever work a problem such as 39 + 83 - 39 from left to right, even though doing so would give the right answer.

If there were such a rule, there would not be so much debate, among professionals, about what 10 ÷ 2 × 5 equals. Rick Norwood (talk) 21:34, 22 April 2023 (UTC)

Wikipedia defines a 'reliable source' at WP:RS. It does not mean what you personally think is a book written by a 'reliable professional mathematician'. Many books support this as you admit above, including the one I cited. We can't overrule this based on your own citation-free personal judgment. MrOllie (talk) 21:39, 22 April 2023 (UTC)

Per Wikipedia policy, college level textbooks are considered reliable sources, but lower level texts (high school, elementary school) are usually not.

See this essay for more info.

Generally, grade school texts are not reliable sources, including the one you cited.

Looking at the bigger picture, "the great thing about standards is that we can have so many of them." There are many "rules" for the order of operations, but there is no universally accepted "standard set of rules". That's probably the main takeaway our readers should get from this article. Mr. Swordfish (talk) 22:03, 22 April 2023 (UTC)

I'm happy we can agree. Grade-school books are not generally considered reliable sources. MrOllie, please cite one book written by a mathematician for adults that has any such rule. Rick Norwood (talk) 23:07, 22 April 2023 (UTC)

Morino, L. (2021). Mathematics and mechanics -- The Interplay. Volume I, The basics. Berlin, Germany: Springer. ISBN 978-3-662-63207-9. OCLC 1257549310. Page 31. It mentions that other systems hold in other places, but the 'prevailing rule in the United States' is left-to-right for operations at the same level. MrOllie (talk) 03:10, 23 April 2023 (UTC)

It is true that in the US, children are taught this rule. But it is a rule for children, not a rule of mathematics. The children in the US who are taught this rule will, if they go to college, learn the real rules, which are the commutative, associative, and distributive laws. Theorems following from these laws say you can add terms in any order and multiply factors in any order. Rick Norwood (talk) 10:11, 23 April 2023 (UTC)

You asked for a source, and I provided it. If you're just going to discount it based on your personal disagreement (again), why did you waste my time like that? MrOllie (talk) 11:37, 23 April 2023 (UTC)

The issue here is that the source says that it is a regional system, but it is not generally applicable. So presenting it as a general rule would misrepresent the source material. Mr. Swordfish (talk) 13:21, 23 April 2023 (UTC)

"about what 10 ÷ 2 × 5 equals" cool. It's a prove that 25 = 1 is true ... 2001:9E8:2447:7E00:A89A:EA00:A03E:9B76 (talk) 18:38, 1 May 2023 (UTC)

Explanatory sentence in Definition section

The first sentence after the four-point list in the definition section was recently removed. It said:

This means that if, in a mathematical expression, a subexpression appears between two operators, the operator that is higher in the above list should be applied first.

It was removed because it could be read to imply that parentheses are an "operator", which they are not.

However, it might still be appropriate to include an explanatory sentence after the four-point list. I'd suggest the following, which does not imply that parentheses are an operator (to my reading at least):

This means that to evaluate an expression, one first evaluates any sub-expression inside the parentheses, working inside to outside if there is more than one set. Whether inside parenthesis or not, the operator that is higher in the above list should be applied first.

Comments? Mr. Swordfish (talk) 17:30, 2 May 2023 (UTC)

Ok for me (the remover of the sentence). I'd suggest to omit "the" before "parantheses", to indicate that several (pairs) of them may occur. And we should keep in mind that the last sentence is to be taken with a grain of salt, since some operators come with their own particular methods to determine their operands (such as root and exponent). - Jochen Burghardt (talk) 19:09, 2 May 2023 (UTC)

Internet memes

I recently added a section "In popular culture" discussing internet memes with ambiguous mathematical expressions. It was reverted because "it was already covered" elsewhere. But the current coverage is buried in the middle of a subsection, doesn't provide much coverage, and cites a source that is self-published.

I don't have any stats to back this up, but my sense is that a large amount of traffic to this page is driven by these internet memes. Since our job is to serve our audience we should provide a more prominent treatment of ambiguous expressions.

I hope that the other editors will consider the since reverted section and think about how to incorporate it into the article. Mr. Swordfish (talk) 15:56, 6 July 2023 (UTC)

Everyone who is lead to our page by an internet meme will find sufficient explanation here. It can't be the purpose of Wikipedia to list the sites that present such memes, or to comment on each particular example that is around. - Jochen Burghardt (talk) 16:10, 6 July 2023 (UTC)

I have added a single sentence in the lead for mentioning the memes. No further explanation is needed, that is not already in the article. D.Lazard (talk) 16:33, 6 July 2023 (UTC)

Thank you. This is an improvement to the article. The new sentence says:

Internet memes sometimes exploit ignorance of the order of operations by writing ambiguous formulas that cause disputes and increase web traffic.

This seems to raise the question of what particular "ignorance of the order of operations" is being exploited. My reading of the article and (most) of the cited sources is that the thing a lot of folks get wrong is the notion that there is one - and only one - truly correct way to parse an expression, hence the "spectacularly vitriolic" arguments. The salient fact is that mathematics is a human language and as with any human language there is the possibility for ambiguous statements. While there are multiple sets of rules (PEMDAS, BEDMAS, chain input, right-to-left, left-to-right, etc.) there is no one universal standard that is "correct" with all the others being "wrong". We have sources that say this, but for some reason the article does not. I think it should. Mr. Swordfish (talk) 18:11, 6 July 2023 (UTC)

The article does say this, explicitly, right above the new sentence. Rick Norwood (talk) 10:38, 7 July 2023 (UTC)

It does? I don't see it. The preceding paragraph is mostly about the use of parentheses.

Later in the article, it becomes abundantly clear that there is no one universal standard that is "correct" once the reader observes that calculators and computer languages have a wide variety of ways to parse and interpret expressions, but as far as I can tell that concept is not expressed in the introduction.

My reading of the introduction is that it implies that there is some universally applicable standard, which contradicts what our sources say. Mr. Swordfish (talk) 12:35, 7 July 2023 (UTC)

The first paragraph begins: "In some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division,... ." In other words, some academic literature (but not all academic literature) uses a different rule from the more common rule, stated above, and multiplication and division, like addition and subtraction, have equal priority. If that still leaves you in doubt, the next paragraph begins "This ambiguity... ." If a rule is ambiguous, then there cannot be one universal standard.Rick Norwood (talk) 12:47, 7 July 2023 (UTC)

Are we looking at the same article? In the version I see, the first two sentences are:

In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.

For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.

This could be read to imply that there is one set of rules for the order of operations that applies to all of mathematics and "most" computer languages. I think we could be clearer. Not necessarily in the first two sentences, but later when discussing ambiguities, which I think deserves its own section rather than a subsection under "Special Cases".

That said, we do need to be careful not to imply "there are no rules, anything goes" since most published works using mathematics are careful not to use ambiguous expressions. Other than strict left-to-right or right-to-left parsing the common conventions will all produce the same result if ambiguities are avoided. Mr. Swordfish (talk) 21:37, 7 July 2023 (UTC)

I see your point. We were looking at different parts of the article. I was looking at the Mixed Division and Multiplication section. I've added a sentence to the introduction which I hope resolves the problem. Rick Norwood (talk) 09:57, 8 July 2023 (UTC)

Thanks. Your edit is an improvement to the article. I've added an internal link to the subsection #Mixed_division_and_multiplication. Mr. Swordfish (talk) 15:14, 8 July 2023 (UTC)

Internet memes 2

Apologies in advance for being a nudge here, but I'm now wondering whether those ambiguous mathematical expressions so often posted on social media are actually memes. I used that term here on the talk page as a kind of shorthand, but did not use it in my proposed edit (https://wiki.riteme.site/w/index.php?title=Order_of_operations&oldid=1163541119#In_popular_culture).

So, are they memes? If so, then clearly they are internet memes and we can call them that. But after reading the definition of meme, I'm not convinced they are, and putting my wikipedia editor hat on, we'd need some source saying that these things are in fact memes regardless of whatever conclusion I might draw from the definitions.

So, anybody got a cite that calls these things "memes"? If not, we should change the language. Mr. Swordfish (talk) 23:03, 9 July 2023 (UTC)

They are one question quizzes. -- Valjean (talk) (PING me) 01:23, 10 July 2023 (UTC)

There are a number of terms that we could use to describe them, "one question quizzes" "social media posts" etc. But I now think we are on solid ground calling them internet memes since one of the cited sources is the website "Know Your Memes" i.e. it is listed as an example of memes.. The Slate article does not call them "Memes" which led me to question whether we had sufficient sourcing. Mr. Swordfish (talk) 12:44, 10 July 2023 (UTC)

I'm not sure that just one source justifies calling it a "meme". They refer to the phenomenon as a "math problem".

It has no relation to the idea of a meme: "A meme is an idea, behavior, or style that spreads by means of imitation from person to person within a culture and often carries symbolic meaning representing a particular phenomenon or theme. A meme acts as a unit for carrying cultural ideas, symbols, or practices, that can be transmitted from one mind to another through writing, speech, gestures, rituals, or other imitable phenomena with a mimicked theme." The only similarity is that it's shared on the internet.

A workaround could be that we first established this as an Internet meme and included it in that article. If multiple sources justified that, then we could call it an "internet meme" here. -- Valjean (talk) (PING me) 16:04, 10 July 2023 (UTC)

"Parenthetic subexpressions"

Why do we use this phrase in the Definition section?

I've looked at every single source cited in the article, and except for the non-US sources that use "Brackets" and the Wolfram cite that uses "Parenthesization" every single cite simply uses the word "Parentheses". And the Wolfram cite links their word to their article on "Parentheses.

So, what's up with us making up our own nomenclature? If there's reliable source using the phrase "Parenthetic subexpressions" we need to cite it. Otherwise, it seems fairly clear to just repeat the language that all the cited sources use. Other opinions? Mr. Swordfish (talk) 00:04, 18 August 2023 (UTC)

My fault ([1]). I wanted to emphasize that parantheses are not operators. What about "Expressions in parentheses or brackets"? - Jochen Burghardt (talk) 08:21, 18 August 2023 (UTC)

The overwhelming majority of cites say something like this:

1. Parentheses,
2. Exponents,
3. Multiplication and Division, and
4. Addition and Subtraction

What's the issue with simply repeating the verbiage in the cited sources? (There aren't any copyright issues in play.) Mr. Swordfish (talk) 12:54, 18 August 2023 (UTC)

 Done, since you insisted.

My point was that exponentiation, multiplication, division, addition, and subtraction are operators, so they are evaluated on their arguments, by applying their respective algorithm. In contrast, parantheses are not operators, so there is no algorithm associated with parantheses that could be used to evaluate them on their "arguments"; what is evaluated is the subexpression inside them. - Jochen Burghardt (talk) 13:12, 18 August 2023 (UTC)

By and large, cites that confuse parentheses with operators are not written by mathematicians, but are written by authors of grade school textbooks. Grade school textbooks in every area of human knowledge are filled with misinformation. It seems there is, on this page, a majority who insist this article contain misinformation. To point out just one example, the article states: "This means that to evaluate an expression, one first evaluates any sub-expression inside parentheses, working inside to outside if there is more than one set." Please follow this instruction and work inside to outside to evaluate the following expression: 2(x+y). Rick Norwood (talk) 12:49, 19 August 2023 (UTC)

Do any of our cited sources "confuse parentheses with operators"? I'm not seeing any. Most don't even use the term operator and the ones that do are careful to distinguish between parenthesis (or brackets) and the operators. Perhaps you could cite an example of what you are claiming?

Similarly, this article does not confuse parentheses with operators. In common language, applying parentheses is an operation in the sense that it is a process, procedure, system, method, or many of other synonyms for the word operation. While there is a mathematical definition of operation, that is not how the word is being used here. Mr. Swordfish (talk) 17:19, 19 August 2023 (UTC)

@Rick Norwood: Usually, only ground expressions can be evaluated. Expressions containing variables can at best be rewritten into some canonical normal form. Mentioning this would, however, be inadequate for a school-level article. - Jochen Burghardt (talk) 18:02, 19 August 2023 (UTC)

Mr. Swordfish asks "Do any of our cited sources "confuse parentheses with operators"?" But earlier he says "Otherwise, it seems fairly clear to just repeat the language that all the cited sources use." So, if the article, when it confuses parentheses and operators, is repeating cited sources, then the cited sources have the same confusion. And if the cited sources do not have the same confusion, then the article should not confuse operators and symbols of grouping.

Jochen Burghardt agrees that the word "evaluated" does not apply to all uses of the order of operations. But claims that to mention this would be "inadequate for a school-level article." The idea that false statements are better than true statements is fairly generally believed by authors of grade-school textbooks in all subjects. If accurate writing is "inadequate", I gather that implies that false statements are "adequate".

Apparently this article will continue to include the highly misleading first two sentences under "Definition". But the advantage of saying things in Wikipedia that students will have to unlearn if they go to college escapes me. Rick Norwood (talk) 10:51, 20 August 2023 (UTC)

Ok. So you don't have any examples of sources we cite that "confuse parentheses with operators".

Thanks for clarifying. Mr. Swordfish (talk) 12:22, 20 August 2023 (UTC)

This is why I have so much trouble when I try to be logical. Mr. Swordfish says "So you don't have any examples of sources we cite that "confuse parentheses with operators"." Where does he get that? Certainly not from anything I said. Rick Norwood (talk) 12:56, 20 August 2023 (UTC)

If you have examples to support your assertions, present them. Otherwise, it's just your opinion, or original research. It's that simple. Mr. Swordfish (talk) 21:02, 20 August 2023 (UTC)

BODMAS, BIDMAS, and geographical distribution

We have solid sourcing that PEMDAS is used in the US and France, and also that BEDMAS is used in Canada and the UK.

We don't have sourcing for the geographical distribution of BODMAS or BIDMAS, so I propose simply writing around the gap instead of making specific claims that are not supported by cites.

Also, the "O" in BODMAS is sometimes said to stand for "Order" (an archaic word for exponentiation), "Of", in addition to the Operations that we state in the text. Seems like we should include those alternatives.

Per the above, I suggest the following text for the third bullet point:

• Other English-speaking countries may use BODMAS meaning Brackets, Operations, Division/Multiplication, Addition/Subtraction. Sometimes the O is expanded as "Of" or "Order" (i.e. powers/exponents or roots). BIDMAS is also used, standing for Brackets, Indices, Division/Multiplication, Addition/Subtraction.

New references are bare urls for now - we can format them in a better manner if this change is accepted. Mr. Swordfish (talk) 14:31, 23 August 2023 (UTC)

UPDATE:

Some sources for geographical distribution:

https://files.eric.ed.gov/fulltext/EJ1148460.pdf

The PEMDAS is an acronym or mnemonic for the order of operations that stands for Parenthesis, Exponents, Multiplication, Division, Addition and Subtraction. This acronym is widely used in the United States of America. Meanwhile, in other countries such as United Kingdom and Canada, the acronyms used are BODMAS (Brackets, Order, Division, Multiplication, Addition and Subtraction) and BIDMAS (Brackets, Indices, Division, Multiplication, Addition and Subtraction).

https://link.springer.com/article/10.1007/s10649-017-9789-9

We further learned that PEMDAS is often used in Francophone Canada.

I don't know that it is this article's purpose to present an exhaustive list of which acronym is used in each geographic area that was once part of the UK, so maybe we should just de-emphasize the geography part. Mr. Swordfish (talk) 16:50, 23 August 2023 (UTC)

I'll only point out (again) that the Common Core standards do not use PEMDAS. I suppose it needs to be here, since most US grade schools ignore the Common Core standards, even when their state requires them. I agree with Mr. Swordfish about shortening this section. Rick Norwood (talk) 10:24, 24 August 2023 (UTC)

A brief summary of how order of operations is treated by the common core standards would probably be an improvement to this article. Looking at the Common_Core#Mathematics_standards section, I'm at a loss as to how one would briefly summarize the coverage of the topic at hand.

The cites are all broken links, so I can't examine the source material. I'd be happy to look at any references to see what they say. Mr. Swordfish (talk) 22:31, 24 August 2023 (UTC)

References

Precedence

The article uses the term "precedence" without defining it. There are almost 20 incoming redirects that have "precedence" in their title, and many others that use the pipe [[Order of operations|precedence]]. In particular, it is almost impossible for a beginner to know which of addition and multiplication has the higest precedence.

So, a definition of precedence must be given in the lead, and this must be expanded in a specific section. Someone is willing to do that? D.Lazard (talk) 16:42, 23 August 2023 (UTC)

My reading is that the meaning is clear from context:

In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.

For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition...

That is, procedures with a higher precedence are performed before those with lower precedence.

Maybe we could change the second sentence to say

For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition (procedures with a higher precedence are performed before those with lower precedence) ...

But I'm not convinced it's an improvement. Perhaps implement it as an explanatory note?

For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition... Mr. Swordfish (talk) 19:00, 23 August 2023 (UTC)

Again, I agree with Mr. Swordfish. There is a Simple English Wikipedia for people who do not know the meaning of common words such as "precedence". Rick Norwood (talk) 10:28, 24 August 2023 (UTC)

In this article, precedence is not a common word, but a mathematical term. See, among many examples Operator precedence in C. D.Lazard (talk) 11:01, 24 August 2023 (UTC)

This is getting really strange. Mr. Swordfish says that in this article it is ok to use the dictionary meaning of operation, which has a special mathematical meaning. D. Lazard says it is not ok to use the common word precedence, which means "which comes first", and is not used here to mean anything else but "which comes first". Rick Norwood (talk) 10:06, 25 August 2023 (UTC)

If it would be the common meaning that is used here, there were not almost 20 incoming redirects with "precedence" in their title (including Precedence (mathematics), and many other incoming links that are piped to "precedence". This article is also for people who follow these links. So, this is an issue that must be fixed, and I'll boldly try to fix it. D.Lazard (talk) 10:35, 25 August 2023 (UTC)

I don't think the most recent edit quite gets it right with regard to computer languages. It says:

The rank of an operator is called its precedence, and an operation with a higher (operator) precedence must be performed before operations with lower precedence. Operations with the same precedence are generally performed from left to right, although some programming languages adopt a different convention.

Most (perhaps almost all) computer languages follow the conventional order, but not all do. And some languages like APL and Smalltalk simply apply a strict left-to-right or right-to-left precedence. It's not just operators with the same precedence that are evaluated in a different order that the norm, all operators are applied in the order they are written regardless of precedence.

The second sentence is half right in that operators that do not obey the associative rule (e.g. division and exponentiation) are applied left-to-right by some languages and right-to-left by others.

I'll suggest:

The rank of an operator is called its precedence, and an operation with a higher (operator) precedence is normally performed before operations with lower precedence. Operations with the same precedence are generally performed from left to right. Some programming languages and calculators adopt a different convention.

By splitting the second sentence, it implies that the third sentence applies to both preceding statements. Mr. Swordfish (talk) 13:38, 25 August 2023 (UTC)

I agree. D.Lazard (talk) 14:09, 25 August 2023 (UTC)

References

"Damning with faint praise"

The opening sentence of the Programming languages section states:

Some programming languages use precedence levels that conform to the order commonly used in mathematics...

and while this is correct, it could be misleading. My experience is that nearly every programming language uses the "standard" operator precedence (with the usual caveats above) so stating it as "Some" implies that it is not the norm, or even a majority.

Looking at this list of most popular programming languages the only exceptions that jump out as exceptions are Lisp and Haskell, which are pretty far down on the list, and html which doesn't really do math at all. Perhaps there are others since I'm not familiar with all of them, but all the more common ones use the standard operator precedence (to the extent that there is one - i.e. grouping symbols, exponents, multiplication/division, and addition/subtraction in that order; beyond that all bets are off)

I don't think it would be original research to look at that list and say "Most commonly used programming languages..." instead of "Some programming languages..." but some may argue that point. Other suggestions for how to word this to avoid possible mis-representation? Or a good cite for this change? Mr. Swordfish (talk) 13:50, 30 August 2023 (UTC)

A side remark: Lisp is not an exception: it uses a purely functional notation and does not need any precedence rule. A specificity of lisp is the placement of parentheses in functional notation: it use the notation (f x) instead of f(x). D.Lazard (talk) 14:07, 30 August 2023 (UTC)

I've never used Lisp (never had the patience to count that many parentheses), but my understanding is that it doesn't use infix notation at all. So, maybe it's an exception in the same sense that html is. Or not. It's certainly different. Mr. Swordfish (talk) 15:06, 30 August 2023 (UTC)

Henderson's Encyclopedia of Computer Science and Technology gives the standard precedence list and says it is used by "most languages". I believe Haskell does have precedence as well, though it is a bit confused by the way you can use the same operator in both infix and prefix versions. D.Lazard is correct, prefix notation in Lisp gets by without any such rules. Smalltalk is the only true counterexample I can think of. MrOllie (talk) 14:13, 30 August 2023 (UTC)

Thanks for the pointer to Henderson's Encyclopedia of Computer Science and Technology. I'll work up a cite and change "Some" to "Most".

BTW, The second programming language that I learned (after Fortran) was APL, which uses strict right-to-left evaluation. I'm not aware of any others, but there probably are although undoubtedly obscure. Mr. Swordfish (talk) 15:19, 30 August 2023 (UTC)

Infix notation

The second sentence of this article is:

These rules are meaningful only when infix notation is used. When functional or Polish notation are used for all operations, the order of operations results from the notation itself.

This is certainly correct, but might be too much this early in the article. My sense is that most of our readers are not familiar with the concepts of functional or polish notation, and those systems are not what this article is about. Granted, there are links for the reader to click, but my reading (assuming the perspective of a non-mathematician) is that this sentence is a distraction from the main thrust of the article.

I'd suggest simply moving this sentence to later in the article, perhaps just a paragraph or two. Alternatively, provide a parenthetical example so the reader doesn't have to click on the link to find out that infix notation is just the familiar way of writing mathematical expressions that they have come to know and love:

These rules are meaningful only when infix notation (e.g. 3 x 4 + 5) is used. When functional or Polish notation are used for all operations, the order of operations results from the notation itself.

Other opinions? Mr. Swordfish (talk) 15:28, 2 September 2023 (UTC)

Agree. I'd suggest to move the sentence to the very end of the lead, and to change "when infix notation is used" to "when the usual notation (called infix notation) is used". Adding an example for infix isn't useful, except when it is contrasted with (e.g.) Polish notation; so we could possible add a sentence like "For example, the infix expressions 3 × 4 + 5 and 3 × (4 + 5) are written as + × 3 4 5 and × 3 + 4 5 in Polish notation, respectively.". I'm afraid, however, that such a sentence won't be understood without additional (and then distracting) explanations. Maybe, reverse Polish notation is easier to explain; in fact is has been employed by HP calculators, so it may be known better. - Jochen Burghardt (talk) 15:54, 2 September 2023 (UTC)

Agree that if we move the sentence to the end of the intro section there is no need for an example. Also agree that providing examples of other notations here would just cause confusion for most of our audience. The curious can click the links to read about the alternative notations. Mr. Swordfish (talk) 18:39, 2 September 2023 (UTC)

I do not object to the new position of the paragraph on alternative notations. However:

• "These rules are" may be unclear after the paragraph on memes. This may be clarified either by replacing "These rules are" by "The order of operations is" or by moving the paragraph before the paragraph on memes. Maybe, the best choice is to do both changes.
• The reason for which I placed this paragrph near the beginning, was to emphasize that there is no mathematical concept here, but only notational convention. It may be useful to clarify this near the beginning of the lead.

I am not sure enough of the best choice for doing these changes myself. So, ... D.Lazard (talk) 21:02, 2 September 2023 (UTC)

Swapping the last two paragraphs does seem to clarify things, so I've gone ahead and implemented that change. That edit seems sufficient, but I can't say I'd object to the other. Mr. Swordfish (talk) 23:05, 2 September 2023 (UTC)

Programming languages

Logical OR and logical AND are non-associative and therefore should have equal precedence. Darcourse (talk) 13:32, 31 January 2024 (UTC)

Logical AND typically binds tighter than logical OR, because people like to write expressions like A && B || C && D and have that mean (A && B) || (C && D). It's possible there's some programming language where this doesn't hold, but it would be mildly practically inconvenient. –jacobolus (t) 15:34, 31 January 2024 (UTC)

Mixing of the rule, the implementation of the rules within different areas (calculator, programming languages)

The sentence "Calculators generally perform operations with the same precedence from left to right,[1] but some programming languages and calculators adopt different conventions. " does not fit where it is placed. The order of operation should apply on mathematical rules in general and not what calculators do in general. This is very confusing because by reading this, I only care on the rules and not what calculators do. Also, in this sentence you mix calculators (what the do most), rules within programming languages which does only explain, an implementation of the rules above. What I can not read, if the rules from left to right by the order of operation is a general rule. So this article for me explains nothing.

It would be much better to have a own section for *calculator* and what the do most, then an extra section for IT and maybe which language rules per default implements different. — Preceding unsigned comment added by Goldnas (talk • contribs) 11:24, 27 January 2024 (UTC)

The rulset itself, the implementation of the rulsets (as described in the article) are different things. Isn't it? Goldnas (talk) 11:26, 27 January 2024 (UTC)

I do not understand your concerns. Indeed, this article is about rules for interpreting formulas in view of doing the implied computation. The considered formulas consist of sequences of numbers (or variables representing them) and arithmetic operators that can be read by a human as well by a computer or a calculator. These rules are conventions, which means that human and computers can use different rules, and, depending of the context, different rules may be used. This is what is said in the article. The implementation of such rules is a very different thing it consist to write a program that follows the rules for interpreting formula. This is usually called an interpreter or a compiler, and it is not the subject of the article. D.Lazard (talk) 12:35, 27 January 2024 (UTC)

I want to see general

• DEFINITION RULE which needs to be done if we have DIFFERENT precedence
• DEFINITION RULE which needs to be executed if operation on SAME precedence

What I do not want to see here, if some calculators or same programming langauge IMPLEMENTATION OF Rule in calculators, in programming language. There is a own section for calculators and for programming language. But the general rule is not written down. The DEFINITION of equal precedence is mixed with IMPLEMENTATION in calculators and programming langauge.

It does not matter if humans or calculators read rules somehow. It is about the definition of rules.

Therefor the current article is not well written it is unclear and if you would refere to this article it would raise more questions like give answers. E.g.

If you refere to this article, some people might say: Yes, but there is not clear definition. Read the sentence:

"Calculators generally perform operations with the same precedence from left to right"

It does not say "Calculators always perform operations with the same precedence from left to right" which means to me, that there is no clear rule because not all calculators do the right thing. So this means I am right If I calculator 60/5*(2+1) I can do the result of 4.

You cannot prove that I am wrong and this article also prove nothing. There no sentence of the rule of evaluate operations on equal precedence, there is only a example of how some calculators and programming languages while it should be the sentence:

On higher ranked operations, evaluate from high to low, on equal rankted operations, evaluate from left to right. That would be a rule which I could implement in any programming language even on those who are not invented yet. Any developer would reject your definition and would ask the same question: How should the computer work if the operations ranked equally? And you would respond: from left to right. So lets write the article clean. The implemtation of rules is in the section available anyhow. Why write same things twice? Makes no sense at all. Goldnas (talk) 20:38, 3 February 2024 (UTC)

The convention for precedence varies from one context to another. The conventions in grade-school mnemonic mantras, pure-math papers, electronic calculators, and various programming languages all differ in various ways. –jacobolus (t) 21:17, 3 February 2024 (UTC)

No, this is not a convention thing. There are rules to execute. I could not implement the ruleset because there is no fully ruleset. And this is an easy one.

60/5*(2+1) evalutes

60/5*3 and then you have (as you can see) 2 opertion on the same level. It should be 100% clear what to do with operations on same level. This rule which is important is not described in the article. And not, this rules do not change, no matter if it is grade-school or pure-math papers. Whatever. The rule for what needs to be done on same level is missing here. Goldnas (talk) 21:37, 3 February 2024 (UTC)

The "rules" (i.e. conventions) are not universal, but vary depending on the context. –jacobolus (t) 21:42, 3 February 2024 (UTC)

Why do you double the statements. The statements about calculator and programming languages are in the other sections as well. Should't we remove it at all? Goldnas (talk) 22:26, 3 February 2024 (UTC)

What is the result of 60/5*3? Goldnas (talk) 22:30, 3 February 2024 (UTC)

I would typically interpret $60/5\cdot 3$ to mean $60/15=4$ in a technical document, and even more so if it were written in terms of variables like $a/bc.$ In a computer program, 60/5*3 instead typically means 12*3 == 36.

The interpretation depends on the context, and an expression like $60/5\cdot 3$ is somewhat ambiguous and often would be better to replace with a more explicit expression such as ${\tfrac {60}{5}}\cdot 3,$ ${\tfrac {60}{5\cdot 3}},$ $60/(5\cdot 3),$ or $(60/5)\cdot 3,$ depending on intention. –jacobolus (t) 22:56, 3 February 2024 (UTC)

But in this case, the section makes also no sense. In this case we would need to remove the sentence. Why? Because in section Calculators and Programming Languages the precedence is described. Goldnas (talk) 19:15, 4 February 2024 (UTC)

The fun part is, that in footnote 11 the write exact the same as I write. So we give a source how it should be done based on the same rule I wrote multiple times in the article but you revert it. Did you read the sources of this article and studied, if there is no contradiction?

Quote:

"BODMAS is an acronyn that serves as a reminder of the order in which operations have to be

carried out when working with equations and formulas:

Brackets pOwers Division Multiplication Addition Subtraction

where division and multiplication have the same priority, and so do addition and subtraction. If

you have several operations of the same priority then you work from left to right."

Sure that we want fo write different in the article compared to its sources? Goldnas (talk) 22:45, 3 February 2024 (UTC)

The sections are already their, my bad. But I still believe it does not fit. It is a repeation of the section below. The rule is missing. Goldnas (talk) 12:11, 27 January 2024 (UTC)

Is a parenthesis a mathematical operation

There is a current version of this article that insists that a parenthesis is a mathematical operation and that PEDMAS is a law of mathematics. A parenthesis is a symbol of grouping and PEDMAS is a mnemonic. No reference is given suggesting otherwise. The person who wants this in the article has restored this claim after my revert with the comment "Take it to the talk page." I would appreciate the opinion of others. Rick Norwood (talk) 19:21, 10 April 2023 (UTC)

Every reference cited by this article treats parenthesis (or brackets or something similar) as part of the "order of operations". Omitting this would mis-represent the source material.

Perhaps you have a source you would like to share that treats order of operations without discussing parenthesis. Mr. Swordfish (talk) 17:06, 12 April 2023 (UTC)

As I mentioned in my revert, you are correct, parentheses are not technically operations, but since they affect that order in which the operations are performed, it becomes necessary to mention them. Furthermore, it is also necessary to know what the priority level of the parentheses is compared to every other operations. So in order to have a standard way to interpret a mathematical formula, a priority level must be given to each operation and to each type of parentheses (or brackets, if one one form of grouping isn't enough). Perhaps, under the definition, instead of just saying "1. parenthesis", it should say something like: "Any operation or series of operations located in the inner most set of parentheses". Then it becomes less confusing since we are talking about executing operations inside the parentheses, and not the parentheses themselves. Dhrm77 (talk) 17:52, 12 April 2023 (UTC)

I suggest 1. Parentheses: any operation or series of operations delimited by parentheses or other symbols of grouping (the order of operations applies also to such grouped operations). This resolves another issue of the current version, namely that it is not said that the order of operations applies also inside parentheses. D.Lazard (talk) 18:32, 12 April 2023 (UTC)

Of course the article has to mention parentheses, and does. My point is that parentheses are a symbol of grouping, not an operation, and that PEDMAS is a mnemonic, not a rule of mathematics. Rick Norwood (talk) 09:52, 13 April 2023 (UTC)

Agree that the definition subsection is not the proper place to introduce the PEMDAS mnemonic. The following three items do not have a parenthetical explanation and I don't think it's necessary to have one for the first item. Do we really need to explain what parentheses are? And if we do, the current version does a poor job of it.

I would support removing the parenthetical explanation from the first item and adding a brief bit of text to the next paragraph explaining that expressions inside of parentheses are evaluated first.

First and recursively. Meaning that if inside parentheses there are another set of parenthesis, that must be interpreted first. Dhrm77 (talk) 15:22, 14 April 2023 (UTC)

Also, the term parentheses is a US-specific term; we should also use the term more commonly used in the rest of the English-speaking world.

I'll suggest the following start to the definition subsection:

The order of operations, which is used throughout mathematics, science, technology and many computer programming languages, is:<ref name="Bronstein_1987"/><ref name="Weisstein_2020_Precedence"/><ref name="Stapel_2020"/>

1. Parentheses or brackets
2. Exponentiation and root extraction
3. Multiplication and Division
4. Addition and Subtraction

This means that to evaluate an expression, one first evaluates any expression inside the parentheses, working outwards if there is more than one set. Otherwise, the operator that is higher in the above list should be applied first.

I think this language also avoids implying that parentheses are an "operator". Mr. Swordfish (talk) 14:53, 13 April 2023 (UTC)

Seeing no objections, I'll make this change. It seems to (at least partially) address issues raised in this thread. Mr. Swordfish (talk) 21:52, 20 April 2023 (UTC)

Well, this has been reverted, with the comment

rv Mr. Swordfish's ongoing insistence that parentheses are an operation, a statement that no professional mathematician I know of believes.

Neither the current version of this article nor the version that was just reverted claims that "parentheses are an operation" so I fail to understand the objection. The proposed revision attempted to address three issues discussed above:

1) PEMDAS is a mnemonic, not a "rule" so it's premature to introduce it in the definition section

2) Operator order is applied both inside and outside the parentheses

3) Parentheses are applied recursively inside to outside if more than one set.

I prefer to resolve this via reaching consensus here on the talk page rather than edit warring. Comments? Mr. Swordfish (talk) 13:14, 21 April 2023 (UTC)

Bemdas or pemdas ... is the correct term ... subtraction is the lowest order of operation ... then addition ... then division then multiplication ... sometimes addition is a higher order of operation than division ... such as in fractions ... 2/3+3/4 ...

one finds the common denominator by multiplying .. then adding then dividing ... making division lower than multiplication at all times ... and Pemdas is technically wrong as each set of brackets has a different name and they are linked under the term brackets ... not parenthesis ..parenthesis are ( ) ... { } are NOT parenthesis ... forgotten their proper name ... but for order of operations all brackets (that is everything in that subset) gets done first ... then exponents ... then multiplication ... and so on ... this once was taught in grade 1 ... back in the 60's ... these days your lucky if it is taught at all ... youre told bemdas and then do it ... so pemdas is technically invalid ... thus why Bemdas is always true ... even the wiki pages have pedmas listed erroneously ... that is NEVER a valid order ... the order of operation doesnt rely on how a calculator does it ... it is a basic math property that what grows goes first what shrinks goes next ... exceptions also relegate higher order operations below lower order operations in exception such as the fraction example I used .. this math rule if you like is as old as Pythagoras ... but back then they didnt have exponents ... and converted multiplication to addition and division to subtraction ... making addition higher than subtraction still ... and thus the order is always bemdas so this order of operations dates back to roughly 537 BC .. to Pythagoras and Thale ... That is the long answer ... you want more involved proof read Thale and Pythagoras ... you only have a few thousand texts to read to get the full explanation ... takes about 3 weeks .. makes a great report for english class btw too

dont forget Parenthesis are ONLY ( ) ... brackets are ( ), { }, [ ] ... the order of brackets is curly first square second and round third ... each have their own meaning and use in the various disciplines of math ... and curly ones are the highest order of them from integration ... square ones are for limits such as from 0 to 4 ... and round ones are just do this stuff before you move on ... square brackets also denote an array ... which is a set of limits defined by rows and columns ...

the mnemonic Bemdas ... is the only accurate one to use ... in all maths even physics and chemistry

as for source grade 1 math 1969 and 1970 ... and further back Thale ... and Pythagoras ... it dates back even further but it is hazy on whom derived it originally ... Pythagoras is the true father of geometry but Euclid got the title ... just like tesla created radio BUT marconi patented it ... then lost it to Tesla 2607:FEA8:BD22:8900:F522:3964:C719:6913 (talk) 00:43, 17 February 2024 (UTC)

this once was taught in grade 1 ... back in the 60's – this was never taught in the first grade. More like 5th–7th grade. It definitely does not date to Thales or Euclid; modern mathematical notation largely arose in the 17th–19th century, and was initially not very settled. The teaching of formal 'order of operations' rules comes from the 19th century.

According to sources I can find, there were actually a few decades in the ~60s–80s where it was taught substantially less than previously, because it was seen as unnecessary/unhelpful. My impression is that it came back into style with the rise of calculators and computer programming languages, which are inherently much pickier than humans.

parenthesis are ( ) ... { } are NOT parenthesis – in this context the word "parentheses" covers all kinds of grouping punctuation. Often these are ordered from inside out as: round "parentheses" (), square brackets [], curly braces {}, and later sometimes angle brackets etc. However, in technical literature it is also common to just use nested parentheses, reserving other kinds of brackets for other purposes such as real intervals, set-builder notation, bra-ket notation, matrices, the Iverson bracket, and so on. –jacobolus (t) 01:48, 17 February 2024 (UTC)

I still think parentheses are a separate topic, which are not clearly explained by an instruction to "do them" first. You don't "do" parentheses. But there is another comment above that I'm curious about: "the term more commonly used in the rest of the English-speaking world". What term is that?

In the US, we talk about (parentheses), [brackets], and {braces} all of which are sometimes used as symbols of grouping and all of which have other uses. Parentheses are used both for points (a,b) and for open intervals (a,b) and there is no way to tell which use is intended except from context. Brackets are use for closed intervals [a,b]. Braces are use for sets {a,b} and an open brace is used for a system of simultaneous equations. (Parenthetic aside, off topic: My students are no longer taught much vocabulary in the US K-12 system, so I now have to say, instead of "brackets", "square brackets" and instead of "braces", "curly braces", for my students to know what I'm talking about.

Wikipedia should have an article titled "Symbols of grouping". I'll think about creating one. Rick Norwood (talk) 10:41, 14 April 2023 (UTC)

See https://editorsmanual.com/articles/brackets-british-vs-american/ or parentheses for the answer to your question. Mr. Swordfish (talk) 14:05, 14 April 2023 (UTC)