Talk:Order of operations/Archive 4

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://en.wikipedia.org/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)
 * 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://en.wikipedia.org/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)