Talk:Order of operations/Archive 2

Wrong
Almost all descriptions of the order of operations are flatly wrong, including the one here. To give just one example, in 2 + 3 + 4*5 it is perfectly ok to add the 2 and 3 before you multiply the 4 and 5. In fact, the very phrase "order of operations" is misleading. What this article is really about is understood parentheses, since the operations of arithmetic are all binary operations. Small wonder that non-mathematicians have trouble understand algebra! Rick Norwood 14:18, 7 May 2006 (UTC)


 * Do you have a source? It looks like that fails the Google Test:
 * Results 1 - 10 of about 23 for "understood parentheses".
 * Results 1 - 10 of about 631,000 for "order of operations". capitalist 02:29, 8 May 2006 (UTC)

Yes, as I said, almost all. In this case 631,000 to 23. All professional mathematicians know better, but few spend any time fighting the battle to improve elementary education. It doesn't do any good, and annoys the education majors. Rick Norwood 19:26, 8 May 2006 (UTC)


 * Well if the goal is to improve communication between the 23 and the 631,000, wouldn't it seem easier to convince the smaller group to change their terminology instead of trying to get the rest of the planet to go along with the 23? That would be the quickest way to get everyone on the same sheet of music.  Or are there factual issues beyond just the terminology?   In other words, if the 631,000 build computers using the information in this article, will they not work as well as computers built by the 23?  If that's the case then the 631,000 must change in favor of the 23.  If it's just terminology then I think the reverse is true.  capitalist 04:24, 9 May 2006 (UTC)

All computers and most calculators use the correct heirarchy. The one in this article is the one usually taught, and in a sense, it works, but nobody who understands mathematics would insist, in the problem 1 + 2 + 3*4, that you must do the 3*4 before the 1 + 2. Clearly, you get the same answer either way. Rick Norwood 21:18, 10 May 2006 (UTC)


 * The article doesn't claim that you won't get the same answer. In the first line it states that O.O.O. is a notational convention, not mathematical fact.  It sounds like just a terminology issue.  Given the preponderance of the term "order of operations" over the term "understood parantheses" my guess would be that the latter term is disappearing from the language.  An interesting case of linguistic evolution I suppose, but at any rate I wouldn't recommend trying to stem the tide through a change in the title of the article.  capitalist 03:24, 11 May 2006 (UTC)

I, too, would oppose a title change. I do think the rule can be stated precisely. Just what more a precise statement would take is something I've been working on. Rick Norwood 17:27, 11 May 2006 (UTC)

I have a question what's the score from: 7-4+3*0+1 - because in every country I have been so far it's 4 and only 4. Poland, UK, Ireland, Slovakia. — Preceding unsigned comment added by 2.26.183.238 (talk) 04:23, 10 July 2012 (UTC)


 * The answer is 4. Rick Norwood (talk) 18:42, 10 July 2012 (UTC)

Re: "Almost all descriptions of the order of operations are flatly wrong, including the one here. To give just one example, in 2 + 3 + 4*5 it is perfectly ok to add the 2 and 3 before you multiply the 4 and 5." This is disingenuous. What order of operations rule does is prevent the 3 being added to the 4. It is true that the addition can be done in two stages as described. But this can only be done by somebody who recognises that the 4 and the 5 are factors not units that must be dealt with before the final addition can be completed. In other words, the problem must still be parsed with PEDMAS first.


 * Mathematicians never use PEDMAS any more than physicists ever use Roy G. Biv. Both are mnemonics for beginners.  Mathematicians know that "order or operations" only applies to the two operations that precede and follow a given numerical expression. Rick Norwood (talk) 14:50, 10 September 2012 (UTC)


 * Rick, I think you're laboring under the belief that the word order implies a chronological order, and not a hierarchical one. The order of operations is a hierarchy that allows one to correctly interpret other algebraic properties.


 * Using commutativity as an example: In the expression 2 + 3 + 4 * 5, the 2 and 3 may be interposed, and 3 + 2 + 4 * 5 will give the same result. Similarly, the 3 and the compound term 4 * 5 may be interposed, and 2 + 4 * 5 + 3 will give the same result.  But the 3 and 4 may not be interposed:  2 + 4 + 3 * 5 gives a different and incorrect result.  The commutativity of the + operator doesn't "outrank" the * operator, because * is of a higher order than + (and not necessarily that it must be done first).


 * Similarly, we may associate 3 and 4 * 5 and compute 2 + (3 + 4 * 5), but we may not simply associate 3 and 4 and compute 2 + (3 + 4) * 5, because the * operator binds the 4 more tightly than the + operator does.


 * Fortunately, interpreting the hierarchical order of operations as a chronological order never results in an incorrect result, and so students are often taught that it is a chronological order before they are exposed to more abstract ideas as commutativity and associativity. --Heath 71.62.157.188 (talk) 02:45, 21 January 2013 (UTC)

Everything you have said is true, but in practice, we use the hierarchy to do things chronologically. As you say, the language for students is active rather than abstract to make it easier for beginners to understand. I just wish my children had not been taught false information. Their gradeschool textbook, written by a member of the editorial board of Mathematics Teacher, said 7 - 4 + 1 = 2 because Aunt comes before Sally. I think that now this Wikipedia article has it right. Rick Norwood (talk) 13:25, 21 January 2013 (UTC)

a^b^c priority order?
Is there any good references for the priority order of a^b^c? I recall from some classes long time ago that a^(b^c) is the correct order, but when Excel, Matlab R2011b and my TI-82 Stat calculator shows the opposite, I'm not so sure anymore, and would be pleased to get some trustworthy references. -- Petter Källström, Petter.kallstrom (talk) 12:23, 14 February 2013 (UTC)


 * The rule is arbitrary, but standard. The reasoning is that a^(b^c) cannot generally be simplified, while (a^b)^c = a^(b*c).  My TI-82 gives 2^3^2 = 512 (rather than 64), so it follows the standard rule.  Does yours not?


 * In the sf novel by Robert A. Heinlein, The Number of the Beast, Heinlein incorrectly calculates 6^6^6. Sf writer Larry Niven wrote to correct him, in a letter that has been published, but I forget where.  As for a more authoritative reference, I looked in four books readily to hand and none mentions the subject at all.  Must be out there somewhere.  Rick Norwood (talk) 13:17, 14 February 2013 (UTC)

Roots or logs?
The article lists exponents and roots, multiplication and division, and addition and subtraction. I'm not sure this is right. Subtraction is addition of the additive inverse. Division is multiplication by the multiplicative inverse. But roots are not the inverse of exponents; logarithms are. Roots are exponents where the exponent is a multiplicative invers.do your prenthesese first and bla bla bla you just do stupid dumb crap ha ha ha bye
 * I'm going to ignore your ha ha ha and seriously answer your question. Addition and multiplication have only one inverse because they are commutative.  Exponentiation is not commutative and therefore has two inverses. The root returns the base (given the exponent).  The logarithm returns the exponent (given the base).  Really, neither belong in the order of operations, because they are not written in the binary form a (operation) b, thus the question of order never comes up.  But the various grade-school level books on the subject, written by people with only a superficial understanding of mathematics (according to the National Counsel of Teachers of Mathematics there are no acceptable mathematics textbooks used in America -- the best is from Singapore) roots get into order of operations as reported in the published literature, and Wikipedia is committed to reporting what sources say.  No original research. Rick Norwood (talk) 00:57, 7 May 2013 (UTC)

Factorials
The comments on factorials are unreferenced and either trivial or extremely dubious. Is there any objection to removing them? --128.101.152.132 (talk) 18:27, 20 August 2013 (UTC)
 * I'm a professional mathematician. How do I know that 2*3!=12 but 2^3!=64.  I don't know.  It just makes sense.  There are traditions in mathematics that may never have been written down.  So, I guess I can't stop you from removing them.  But they do make since and they do offer valuable advice for students. Rick Norwood (talk) 00:05, 21 August 2013 (UTC)
 * You are not the only professional mathematician in this conversation. The section in question asserts far more than that Rick Norwood believes 2*3!=12, it asserts that it is an established convention that 2n!=2(n!).  If this were the case, it would be easy to find sources supporting it.  (My personal belief is that no universal convention for resolving 2n! exists, that (2n)! is probably the more common meaning, and that anyone who cares about being understood will use parentheses.) The question of how to interpret 2^3! is a very odd one as mathematics is not, generally speaking, written in ASCII notation, and moreover the discussion in he section is about certain totally unambiguous non-ASCII expressions.  Separately, I don't understand your last sentence. --70.99.180.246 (talk) 01:07, 21 August 2013 (UTC)

Your assumption that all the rules have been written down is interesting. That has not been my experience. In 1900, Hilbert could probably have made rules that every living mathematician would listen to, but I can't think of anyone who has that authority today. Steven Krantz? Not even he, in my opinion. I certainly agree that careful mathematicians use parentheses when there is any possibility of misunderstanding. Rick Norwood (talk) 01:21, 21 August 2013 (UTC)
 * I find the idea that everyone agrees 2n! = 2(n!) but no one has ever written this down to be very strange, compared with the more plausible alternative that it doesn't come up very much and so there has never been a need for a convention to develop. 70.99.180.246 (talk) 02:48, 21 August 2013 (UTC)
 * I agree with the IP editor. In the absence of a reputable source, the article should not assert anything about the interpretation of factorial expressions. Zueignung (talk) 04:23, 21 August 2013 (UTC)

Wolfram alpha treats 2^3! as 2^(3!)=64 that is what I would expect most other mathematical engines to do as ·! is a unary operator and normally unary operators have a higher precedence that binary mathematical operators. --Salix (talk): 06:43, 21 August 2013 (UTC)
 * I like your observation about unary operators having higher precedence than binary, though that does not (usually) apply to unary minus. Actually, when I said "never written down" I overstated the case.  Probably somebody has discussed the issue, maybe in a paper on combinatorics.  What I should have said is that I'm not aware of anyone writing down a rule other than here.  And, while Wikipedia is not supposed to decide such issues, it probably has that effect, at least in some cases. One compromise would be for this article to just report what Wolfram alpha and Texas Instruments in fact do with factorial.  Rick Norwood (talk) 11:48, 21 August 2013 (UTC)
 * I agree that discussion of "in practice" rules for various computational systems would be of interest. 70.99.180.246 (talk) 20:23, 21 August 2013 (UTC)

left to right
From time to time someone edits this pages with information they were taught in high school, that operations on the same level must be performed from left to right. The teachers who teach this, and there are many, no doubt have the best of intentions. But, as all professional mathematicians know, it simply is not true. For example, in the expression 3 - 4 + 5 + 6, there is nothing at all wrong in adding the 5 and the 6 before adding the 3 and the -4. 3 - 4 + 5 + 6 = 3 - 4 + 11 = -1 + 11 = 10. Rick Norwood (talk) 01:16, 13 November 2013 (UTC)
 * For programming languages the order is defined. See for example C operator precedence and Operator associativity partially because computers need to be told how to do everything and also to resolve ambiguous situations like 3 − 4 − 5. Curiously some operators bind right to left, notably the power operator. --User:Salix alba (talk): 06:38, 13 November 2013 (UTC)

PEMDAS is wrong
I also learned PEMDAS in school. However, consider the expression:

4 / 2 * 2

What is the answer? The correct answer is 4. Division should come before multiplication. PEMDAS would lead you to believe that the correct answer is 1.

There may be counter examples out there that I'm unaware of, but if not, it really should be PEDMAS. Please Eliminate Dumb Math Acronyms at School :) —Preceding unsigned comment added by 12.239.58.227 (talk) 02:41, 12 May 2010 (UTC)


 * The article mentions that PEDMAS is wrong. Now if only we could get the grade school textbooks to stop teaching students wrong math.  But that seems impossible. Rick Norwood (talk) 14:06, 12 May 2010 (UTC)


 * I thought the Order of Operations acronym was GEMA or GEMS: Grouping - Exponents - Multiplication & division - Addition & Subtraction/Subtraction & Addition. --DanMat6288 (talk) 00:09, 24 June 2010 (UTC)

GEMA is better than PEDMAS, but it is still wrong. In 3 + 4 + (5 + 6) I can do 3 + 4 before I do 5 + 6. This article states in the first paragraph the actual rule, but no grade school text book I know of does. Rick Norwood (talk) 12:56, 24 June 2010 (UTC)

Unfortunately, PEMDAS or GEMS are simply acronyms to help people remember basic order, they cannot be taken as a hard and fast rule. Students should know the mathematical properties before learning order of operations using an acronym. For example, the example given above 3 + 4 + (5 + 6) is correct, you can add any of those numbers before adding the other, the parentheses are superfluous. The associative property of addition states that the order of the addends does not matter when adding more than three numbers. Without understanding the mathematical properties, PEMDAS and GEMS will only take you so far. — Preceding unsigned comment added by 71.175.67.224 (talk) 20:31, 8 June 2014 (UTC)


 * There seems to be no one standard, unfortunately. The local junior college teaches the GEMS sequence (though without using the acronym).  I.e., multiplication and division are at the same level, so you do them left to right.  Same for addition and subtraction.  But the local GED program teaches PEMDAS, with multiplication taking precedence over division, and addition over subtraction.  So the few GED students who need to take the college entrance diagnostic have to relearn a different system.  — Preceding unsigned comment added by 71.101.39.171 (talk) 01:50, 8 May 2013 (UTC)

The above sentence is not true. PEMDAS tells you to work from left to right so 4 divided by 2 = 2, multiplied by 2 = 4. Simple! — Preceding unsigned comment added by 162.234.64.49 (talk • contribs) 03:45, 27 November 2013

The usual answer is that the above expression should be read as 4 * (1/2) * 2, and then PEDMAS works as intended. Doing the multiplications first gives you (4/2) * 2 = 8/2, or 4 * (2/2) = 8/2, depending on whether you do the first or second multiplication first. This is all already explained in the mnemonics section, with the example given for addition/subtraction rather than multiplication/division. Maybe it would be a good idea to add an example for that too, in case it's not clear that the same argument applies. (Though I agree it would be better to simply change the mnemonic, but that's not Wikipedia's job - it just reports how things are, not how they should be.) Quietbritishjim (talk) 15:27, 27 November 2013 (UTC)

"most textbooks"
People often edit this page based on what they were taught in grade school, but much US primary education has its own rules, which are not used by professionals. One of the causes of the math wars was the unsuccessful attempts by professional mathematicians to get grade schools to teach mathematics the way it is done beyond grade school.

A recent edit claimed, without references, that "most textbooks" teach that implied multiplication has higher precedence than written multiplication and division. I recently graded the AP calculus exam in Kansas City, and students who used that "rule" lost points. 1/2x should, as the article states, be avoided. But when it is used it is read "one half x", that is, 1 divided by 2 and then the quotient multiplied by x.

Another point raised by the same edit: functions are not considered binary operations.

Rick Norwood (talk) 12:28, 8 July 2014 (UTC)


 * I imagine it depends on what you consider a "textbook" to be. At least here in the UK, even university-level books are often called "textbooks", at least in undergraduate, so the edit may have been correct with that interpretation. But I agree that implicit multiplication is more likely to be considered lower precedence than division in schools (not colleges/universities) because of teachers' slave-like adherence to rules like BODMAS as if they were handed down by gods. In contrast, professional mathematicians would feel confident that 1/2x would be interpreted by their peers as 1/(2x) because if they had meant (1/2)x they would have just written x/2. I see implicit multiplication used this way extremely often in the exponential, such as writing the Gaussian PDF as $$\tfrac{1}{\sigma\sqrt{2\pi}} e^{ -(x-\mu)^2/2\sigma^2 } $$, which surely no experienced person would find confusing.


 * Of course, regardless of how obvious it is to those that practise it that this is standard, it needs a citation to go in the article text. Quietbritishjim (talk) 19:52, 10 July 2014 (UTC)

The Gaussian PDF is an interesting example. Because it is a familiar formula, people "understand" what it means. But how about $$2y=x, y=1/2x $$? I see that in class often. The AP calculus committee decided to mark that right if the intent is half of x, wrong if the intent is one over the quantity 2x.Rick Norwood (talk) 21:34, 10 July 2014 (UTC)


 * If x/2 were intended then that's surely what would've been written, whereas 1/2x is a useful decluttering of 1/(2x), and it seems visually clear that the 2 is more closely associated with the x than the 1. If I hadn't seen the Wikipedia page it wouldn't even have occurred to me that another interpretation were possible. I would expect it to normally be written $$\tfrac{1}{2x}$$, but in an exponent (as with the Gaussian) it's better to write fractions in the x/y style because otherwise the denominator is almost the same level as the base of the exponent. I suspect that's why the problem re-emerges, with the different interpretation, in advanced settings. Quietbritishjim (talk) 00:16, 12 July 2014 (UTC)

Quietbritishjim: Maybe in Great Britain "surely" x/2 would be written. In the US, not so. Fractions are taught badly here in grade school and high school, and so I must teach my calculus students that 1/2 * x and x/2 are equal. Until I teach them, many take the derivative of x/2 using the quotient rule! However, the larger point is this. There is no consensus on whether implied multiplication takes precedence over written multiplication, nor is there a consensus over whether a large multiplication "dot" acts as a symbol of grouping. In the absence of consensus, all this article can give are the standard rules, as written in many books, which do not distinguish between the several symbols for multiplication. Blame Leibniz, who introduced "understood multiplication" to avoid dealing with the fact that in England the decimal was a dot on the line while multiplication was a dot above the line, while on the continent it was the other way around.Rick Norwood (talk) 11:53, 12 July 2014 (UTC)

Collective terms
Regarding these three groupings:
 * 1) exponents and roots
 * 2) multiplication and division
 * 3) addition and subtraction

Would anyone know if collective terms exist to express these? Like is there a term meaning "add or subtract" or a term meaning "multiply or divide" or similar? I would think it would be informative to include that in the article but nothing comes to mind. Ranze (talk) 03:14, 29 July 2014 (UTC)


 * Adding and subtracting can be called "combining like terms", and multiplication and division can be called "expanding and reducing", exponents and roots can be called "powers", with roots being fractional powers, but none of these are common enough to be used in the article. As is mentioned in the article, we only need three binary operations: addition, multiplication, and exponentiation.  Subtraction is addition of the opposite, division multiplication by the reciprocal, and roots can be expressed as fractional exponents.  I don't know if this useful information is taught in schools or not. Rick Norwood (talk) 12:06, 29 July 2014 (UTC)

Matlab
A recent edit claimed that Matlab evaluates exponents from left to right instead of top down. This reference says otherwise. http://math.boisestate.edu/~wright/courses/matlab/lab_1/html/lab_1.html "When using stacked exponents, the convention is to work in top down order. So for example, the expression 2.1^4.3^2.1 should be evaluated using parenthesis as 2.1^(4.3^2.1)"

But it may be that different versions of Matlab use different rules. In any case, the edit was unreferenced, so I reverted it pending more detailed information from an expert on Matlab. Rick Norwood (talk) 13:52, 8 October 2014 (UTC)


 * That particular quotation seems to suggest that *user* needs to add parentheses in order for it to be evaluated in the conventional way, which hints that maybe it will be evaluated the other way if you don't explicitly put parentheses in. But looking at the rest of the page, it doesn't say outright either way, and there are not evaluated examples of stacked exponents without them. I think it would be best to find another source. (I don't have Matlab installed to test which is really correct.) Quietbritishjim (talk) 10:21, 9 October 2014 (UTC)

"1/2x is interpreted as 1/(2x) by TI-82"
I can attest that this is also true of the Casio Fx-991MS scientific calculator; it is not interpreted as (1/2)x. If there's a number immediately the left of brackets, it is calculated first. So the answer for "6÷2(1+2)" is 1, rather than 9.

The thing is, I seem to remember that this is also the order of operations that was taught in math class in Ontario, Canada, all through high school. In order for the answer to be 9, it would have been written as "(6÷2)(1+2)". Esn (talk) 09:19, 19 November 2014 (UTC)


 * I (and others) have tried to point out, ever so gently, that $$1/2x = 1/(2x)$$ is a widely held convention; i.e., it is often not the case in scientific literature that multiplication and division have equal precedence. But other editors won't have it. Zueignung (talk) 10:18, 19 November 2014 (UTC)

It is not the case that "other editors won't have it"; it is a case where authorities disagree. There are two conflicting widely held conventions, which is unfortunate, but not something Wikipedia (or anybody else, apparently) can settle. As for what is taught in grade school, all authorities agree that what is taught in grade school is wrong (e.g. you must work from left to right), but the grade school teachers are not about to change just because what they teach is wrong. They teach what they were taught.Rick Norwood (talk) 12:29, 19 November 2014 (UTC)


 * Please provide the names of these authorities. Then please insert them into the article, because so far it is mostly uncited and reads like original reasearch. There are almost no references in the article supporting the description of "the standard order of operations". Zueignung (talk) 18:39, 19 November 2014 (UTC)

The associative law of addition, which I hardly need to cite sources for since it in virtually every textbook, says that (a + b) + c = a + (b + c). Therefore, as all authorities agree, I can work addition problems from right to left rather than from left to right. Nobody disagrees with that.

On the other hand, punch "6/2(1+2)" into different calculators, and you get different answers, which shows that the people who program calculators have different opinions. Rick Norwood (talk) 22:42, 19 November 2014 (UTC)
 * Is this convention consistently different in different geographical locations? Or institutions? Where is the border, precisely? (Also, what is the history of the standard order of operations? Who decided on these rules? Was there a split on this very issue right from the start?) Esn (talk) 12:32, 2 December 2014 (UTC)

I've tried to locate a scholarly treatment of the subject, but all of the many published books on the subject that I have been able to find are written by primary schoolteachers who disagree among themselves and say things that no mathematician would accept, e.g. you must add from left to right.Rick Norwood (talk) 13:07, 2 December 2014 (UTC)

I've never seen anyone use the 1/(2x) convention anywhere in published papers in maths or computer science - it seems to be exclusively something you see at the primary or secondary school level. And it sounds like physics uses the same convention of 1/2x. Rick, can you provide a single example of a journal that does it your way? Because there are already examples cited of journals that do things the other way. Ramseytheory (talk) 12:05, 27 July 2015 (UTC)


 * I'm not sure what you mean by "my way". I only point out that some sources use one convention, some the other.  For example type 1 ÷ 2x into a TI-89, you get (1/2) x. According to esn above, the Casio Fx-991MS returns 1/(2x).  All Wikipedia can do is report that there are two conventions, and suggest people use parentheses if they want to be clear.  Rick Norwood (talk) 12:54, 27 July 2015 (UTC)


 * Right now though, the page gives the strong impression that 1/(2x) is more common in almost every setting and that maybe one or two specific journals do things differently. The clause "However, there are examples, including in the published literature..." is a lot weaker than "However, in most of the published literature...". The latter version would be more accurate, and this would be an easy change to make. Ramseytheory (talk) 15:21, 28 July 2015 (UTC)

I agree with you, and I think most mathematicians use that convention. In the past, though, attempts to say so have annoyed people who grew up with the other convention. I suggest a text that does not preference either version. Rick Norwood (talk) 11:55, 29 July 2015 (UTC)


 * I've just noticed that what I wrote above is slightly ambiguous - to clarify, my position is that in academia, 1/2x is generally taken to mean 1/(2x) rather than (1/2)x, and so you wouldn't normally write 1/(2x). Hopefully that meaning came through, sorry if it didn't. In any case, I don't think replacing "However, there are examples, including in published literature, where implied multiplication..." by "However, in most of the published literature, implied multiplication..." does preference either version. It doesn't say the version these people grew up with is wrong, just that it's rarely used in academia. Would it be OK for me to make that change? Ramseytheory (talk) 12:52, 29 July 2015 (UTC)

Not unless you have some evidence for "rarely used in academia". Rick Norwood (talk) 13:48, 29 July 2015 (UTC)


 * This is silly: none of this stuff is referenced, and neither the current version nor the proposed version has any proper evidence for it. (And probably no actual evidence exists, i.e., no one bothers to write proper secondary sources about this kind of thing, and for good reason.)  I like Ramseytheory's version well enough, possibly also changing "most" for "much".  Also, I think "published literature" is going to be ambiguous or unclear for many readers: most people who come to read an article like this aren't academic mathematicians.  --JBL (talk) 14:16, 29 July 2015 (UTC)


 * Agreed with Joel. Rick, you yourself said above that you tried to find a proper source on academic conventions and came up blank. I tried as well, with the same result. Meanwhile, we've at least got evidence in the existing citations that 1/2x = 1/(2x) is a common convention in academia, if not the most common. Do you have evidence to the same standard for 1/2x = (1/2)x, e.g. some major textbooks or journals that use it? Because if you can't find any then I think it's fair to change the article to say it's rare, and then anyone who can come up with evidence to show that it's common can update the article and cite it.
 * That said, the "right" way of solving this is probably to just look at the 10-20 most recent mathematics submissions to arxiv. I would expect most of them to use 1/2x = (1/2)x, a few of them to either never write inline fractions of the form 1/2x, and none of them to use 1/2x = (1/2)x. I'm guessing this would constitute original research and couldn't be cited, but I'd be happy to do it if you would accept the result. (Joel, good point on "published literature" - how about "academic literature"?) Ramseytheory (talk) 12:20, 30 July 2015 (UTC)
 * That said, the "right" way of solving this is probably to just look at the 10-20 most recent mathematics submissions to arxiv. I would expect most of them to use 1/2x = (1/2)x, a few of them to either never write inline fractions of the form 1/2x, and none of them to use 1/2x = (1/2)x. I'm guessing this would constitute original research and couldn't be cited, but I'd be happy to do it if you would accept the result. (Joel, good point on "published literature" - how about "academic literature"?) Ramseytheory (talk) 12:20, 30 July 2015 (UTC)


 * Probably you should edit the key sentence (about what you would expect to find). I would guess option 2 (avoid the issue entirely) is very, very common.  I agree with "academic literature" as an improvement. --JBL (talk) 13:17, 30 July 2015 (UTC)


 * Probably true, actually - a lot of people prefer to avoid inline fractions entirely, for example. I'll stand by 1/2x = (1/2)x being much rarer than 1/2x = 1/(2x), though. Ramseytheory (talk) 16:57, 30 July 2015 (UTC)

I'm in academia and have been teaching math all my professional life, and had never seen 1/2x = 1/(2x) until the issue came up here. The TI-89, which sets 1/2x = (1/2)x is the most popular academic calculator. And, logically, the meaning of 1/2x should not depend on the symbol we use for division and the symbol we use for multiplication, else we would have to memorize a complicated set of rules for every pair of symbols. Rick Norwood (talk) 15:19, 30 July 2015 (UTC)


 * When one talks about "the academic literature," one is certainly not discussing calculators that are rarely used by anyone beyond the context of a sophomore-level mathematics class. Your last sentence is a total nonsequitur, since this is a question of actual usage in real life.  Ramseytheory, I suggest you go ahead and make the change.  (Though, as I mentioned, I think "much" is more defensible than "most.") --JBL (talk) 15:47, 30 July 2015 (UTC)


 * I've made the "much" version of the change. For what it's worth, I've also been in research for the last 5 years (as a PhD student and postdoc) and my experience agrees with JBL's that mathematicians and computer scientists rarely use calculators. Normally if the numbers are big enough to need one then Mathematica or some other CAS is a better tool... Ramseytheory (talk) 16:57, 30 July 2015 (UTC)

My second sentence, which you dismiss as a "non sequitur", is the main point. The rules should be simple and consistent, not one rule for 1/2x, one rule for 1/2*x, one rule for 1/2(x), one rule for 1÷2x, one rule for 1÷2*x, and one rule for 1÷2(x). Your way looks right to you because you're used to it. The rule I've always used looks right to me. The same is true about the never-ending controversy over whether a ring has a multiplicative identity. What you learn first looks right. Please do not let your personal experience influence your Wikipedia edits. Do not say that much academic literature uses either convention unless you have a source. Otherwise, all you are saying is much of the academic literature you've happened to run across uses this convention. Rick Norwood (talk) 17:01, 30 July 2015 (UTC)

I've changed "much" to "some". If you have evidence for "much" I'll change it back. Note that the convention in the footnote "In both books these expressions are written with the convention that the solidus is evaluated last." is an even stronger claim. Under the solidus evaluated last rule, 2x + 1/2 + 3 means (2x + 1)/(2 + 3). You can see how messy things can get when somebody, even as great a writer as Feinman, makes up their own rules. Rick Norwood (talk) 17:11, 30 July 2015 (UTC)


 * As JBL said, under NPOV it is totally irrelevant what the rules should be since we're talking about what they are - to the extent that the paragraph is normative at all it already suggests that 1/2x = (1/2)x is the "right" way of doing things. A lot of people think that pi should be replaced by tau/2, but pi remains the dominant convention. I'm not doing this because 1/2x = 1/(2x) "looks right to me", I'm doing it because to the best of my recollection I have literally never seen a paper or university-level textbook that uses 1/2x = (1/2)x. And I've now asked three times, but you have still yet to provide a single source showing that even some of the academic literature uses 1/2x = (1/2)x. I'd settle for a major journal's style guide and some textbooks, i.e. the same level of evidence currently present for 1/2x = 1/(2x). If you can't provide this, which should be easy if "much" isn't correct and 1/2x = (1/2)x is dominant in your area, then I'm reverting your change. If you can provide it, then in lieu of an actual study to prove "much" I'll settle for rewording the paragraph to say that some of the literature does one thing and some does the other (with citations for both). Ramseytheory (talk) 09:54, 31 July 2015 (UTC)

Since you cite your credentials, I'll cite mine. Ph.D. in mathematics, member of the Institute for Advanced Study in 1980-81, twenty-eight published papers. All of which is totally irrelevant. Your have had your experiences, I have had mine, our experiences have been different, and are entirely beside the point, since a Wikipedia editor is not allowed to use their own expertise to justify their edit, but must use published sources. It is not up to me to find a published source that says that "much" is wrong. It is up to you to provide a published source that "much" is right. Rick Norwood (talk) 13:16, 31 July 2015 (UTC)


 * There already is some published evidence that "much" is right - Physical Review is a pretty major journal and as such very unlikely to take their submission instructions contrary to widespread practice in their field. AMS reviews had a similar policy, but removed it - presumably since the point of the paragraph was to tell people to in-line formulae to save on printing costs. Two major textbooks using the convention are already cited, and this StackExchange answer points out that Rudin uses it as well (together with some less fundamental books). If it would be helpful I could find many other examples of textbooks using the 1/2x = 1/(2x) convention as well.


 * Moreover, the article as it currently stands implies by omission that most of academia uses 1/2x = (1/2)x. And you haven't provided any evidence even for the far weaker claim that some of academia uses 1/2x = (1/2)x. Even if the evidence for "much" weren't there, all that would mean is that the article should be phrased "some... some..." - which would still require you to put up some evidence of 1/2x = (1/2)x being used.

Ramseytheory (talk) 15:49, 31 July 2015 (UTC)

Here is the evidence you requested.


 * "Multiplication vs. Division and Addition vs. Subtraction. What happens when an expression involves more than operator at the same level of precedence? (For example, more than one multiplication and division or more than one addition and subtraction.)  What would be two possible values for this expression?  2/3*4; The division was performed first (to obtain 2/3), and the result was then multiplied by 4 (to obtain 8/3). Had the multiplication been performed before the division, the result would have been 1/6.  This does not mean that multiplication takes precedence over division, however. In the absence of parentheses, multiplication and division are performed left to right. We say that multiplication and division are left associative."  Operator Precedence Worksheet. https://www.cs.utah.edu/~zachary/isp/.../operprec.html.


 * "Sometimes one rule seems natural, and sometimes another, so people will forget any rule we choose to teach in this area. I've heard from too many students whose texts do "give an example that really puts this rule to the test," but do so by having them evaluate an expression like: 6/2(3) that is too ambiguous for any reasonable mathematician ever to write."  Doctor Peterson, The Math Forum   http://mathforum.org/dr.math/.


 * "division and multiplication have the same priority," Rules of arithmetic - Mathcentre www.mathcentre.ac.uk/resources/uploaded/mc-ty-rules-2009-1.pdf.

I've looked at about a dozen of these on google scholar, avoiding the ones that refer to a particular programming language, and all either say that multiplication and division have the same precedence, or that people disagree, and so there is no fixed rule. Wikipedia should not say that the multiplication before division rule is the one that is most common. Rick Norwood (talk) 13:58, 2 August 2015 (UTC)


 * Rick Norwood, so far, to defend the idea that if one comes across the string "1/2x" in the academic literature one should expect it to mean x/2 and not 1/(2x), you have offered the following defenses:
 * statements about the behavior of certain calculators;
 * personal statements of incredulity; and
 * references that do not discuss the academic literature.
 * This is ... not compelling. --JBL (talk) 21:12, 3 August 2015 (UTC)

Just to be sure we understand each other. You think references on google scholar are not important, but your personal experiences are? Rick Norwood (talk) 22:40, 3 August 2015 (UTC)


 * No, you do not understand me. Perhaps you should go re-read the discussion.  --JBL (talk) 23:23, 3 August 2015 (UTC)

Ok, I've re-read the discussion. In favor of 1/2x = 1/(2x) I find Physical Review, a physics journal. If you want to say that most physics journals use this convention I have no objection. I find a statement that the AMS had this convention but changed it. That is, if anything, evidence that the consensus is shifting. Wolfram Alpha also had this convention but changed it. StackExchange is unrefereed. Anybody can post there. If Rudin used this convention, you should reference Rudin by book and page number. I looked through Real and Complex Analysis and Functional Analysis and did not come across it. I can hardly read whole books looking for it. This is all the evidence I find in favor of 1/2x = 1/(2x). Let me know if I missed any. In favor of 1/2x meaning (1/2)x, I've cited Operator Precedence Worksheet, "In the absence of parentheses, multiplication and division are performed left to right." and Mathcentre in the UK, "division and multiplication have the same priority". I could easily cite more sources. Let me know how many you want. I also cited "The Math Forum", which says "6/2(3) that is too ambiguous for any reasonable mathematician ever to write." but Math Forum is also unrefereed, though it is written by a mathematician. I've published an article on the subject in Mathematics Teacher, a refereed journal, but it is math ed and in any case I hesitate to cite my own publication. My conclusion is that we have a lot of evidence that both conventions are used, some evidence that division and multiplication, like addition and subtraction, have the same priority, and no evidence that "most" sources give multiplication priority over division. Rick Norwood (talk) 12:23, 4 August 2015 (UTC)


 * Rick, are you ever going to add citations to the body of the article in support of your claims? Or are you just going to continue to waste other editors' time by intentionally misconstruing what they say on the talk page? Zueignung (talk) 03:12, 29 August 2015 (UTC)

I added a citation but someone reverted it without explanation. Now another person has added a citation. I'll add yet another. Please show me where I have ever intentionally misconstrued what someone else said. Rick Norwood (talk) 11:57, 29 August 2015 (UTC)
 * Rick, JBL reverted you because the link in the reference you gave was ill-formed and non-functional. Per WP:ROWN, what he should have done instead of reverting is just fix the citation for you. However, you could probably improve your citation style as well (see Template:Citation). ;-) --Matthiaspaul (talk) 23:36, 30 August 2015 (UTC)

Thank you for the information and the suggestion.Rick Norwood (talk) 11:36, 31 August 2015 (UTC)

Implied groupings
This article is badly misleading because it uses P to only mean explicit groupings. But groupings can be both explicit and implicit. Take for example the expression 1/2x. By not using an explicit operator between the 2 and the x, an implicit grouping has been implied.

No sane person would want to write x/2 as 1/2x. It makes no sense to write it that way. On the other hand 1/2*x has no implicit grouping so it can easily mean x/2, even though it is not a nice way to write it...

To ignore implicit groupings simply leads to a generation that cannot do basic algebra. Just try doing vector operations without understanding implicit groupings. It would lead to attempting to divide by vectors and other such nonsense operations.

I will attempt to find some good references to explicit groupings before editing the article.. Bill C. Riemers (talk) 01:48, 26 March 2016 (UTC)

Here one excellent article by a professor from Berkeley that incates PEDMAS does not even apply to expressions like 1/2x because the implied grouping makes it ambiguous. https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html As such this example should be modified to use an implicit operator. Bill C. Riemers (talk) 02:06, 26 March 2016 (UTC)


 * This subject has been discussed here at length above. Scientists tend to agree with you that 1/2x means the reciprocal of 2x.  Mathematicians tend to see it as x/2.  I'm a mathematician.
 * I would argue that once we open to the door to the idea of "implicit grouping", do we then say that 2x^2 means the square of 2x? Certainly not.  So, how to formulate the exact rule for when 2x is explicitly grouped.  Is it grouped in √2x?  Is it grouped in 1/2 x?  How large does the space between the 2 and the x have to be before the implicit grouping is broken?  Is half an em enough to break the implicit grouping?  How about an em?  You see the bag of worms opened up.  Essentially, scientists say "if it looks like implicit grouping to me, it is, and you mathematicians spend too much time worrying about definitions."  Yes, we do worry about definitions, but we have reasons.


 * In any case, this is not the place to debate the subject. Wikipedia uses references and while there are references both ways, the preponderance of the references agree with the article as it stands.  Almost all references agree that 1/2x should be avoided.


 * Rick Norwood (talk) 11:58, 26 March 2016 (UTC)


 * "Mathematicians tend to see it as x/2." This claim continues to be completely false, and you have never produced a single piece of evidence that supports it. --JBL (talk) 14:20, 26 March 2016 (UTC)
 * As soon as you get into higher order mathematics, even vector operations the PEDMAS rules simply fall apart. As such, most mathematicians could care less about the silly PEDMAS rules taught in grade school.108.170.149.169 (talk) 18:31, 26 March 2016 (UTC)


 * The point between implied groupings, is that if there is an implied grouping that is ambiguous then the rules PEDMAS should simply not be applied. If you do not know if the author that wrote 2x^2 intended an implied grouping between the 2 and the x, then you should not proceed to blindly calculate a result, but should instead seek clarification on what they meant.  I would however, argue it is clear in the case of 2x^2 the author means 2(x^2) because exponents are a higher order operation than multiplication.  The ambiguity really only becomes likely with two operations of the same order prescience.108.170.149.169 (talk) 18:32, 26 March 2016 (UTC)

I am certainly not implying that PEDMAS or any of the other silly grade school rules apply. As discussed in an issue of the Notices of the American Mathematical Society last year, all American K-12 math textbooks are hopelessly flawed. I have given a number of references (in the previous section) where the standard hierarchy of operations applies, with addition and subtraction on the bottom level, multiplication and division on the middle level, and exponents and roots on the highest level. I've also pointed out the ambiguity that arises when "implied grouping" is asserted. Others disagree, and I'm under the impression that most mathematicians believe whatever they learned in their first real math course is the right way to do things, and any other way seems wrong to them. (It's the same with beliefs about whether a ring must have an identity.) Since it is all a matter of definition, the important thing is to use parentheses to avoid possible ambiguity. Rick Norwood (talk) 11:58, 27 March 2016 (UTC)

history?
it would be good to have a written history of the development of this process, and why certain operations were chosen for precedence over others, and why left-to-right operation was eliminated. tpk (talk) 15:50, 18 January 2014 (UTC)
 * I would also find it interesting, though the history may by this time be obscure or lost. The order of operations is logical: exponentiation comes before multiplication because an exponent is used to indicate repeated multiplication, multiplication comes before addition because multiplication is used to indicate repeated addition.  I'm not sure there was ever a "left to right" rule, except in certain grade school classes where the teacher, usually not a mathematician, thought it was a good idea. Rick Norwood (talk) 17:06, 18 January 2014 (UTC)


 * I too would like to see a history section discussing the historical evolution and variance over time. The Dutch, French and German WPs have quite a bit of material on this topic already. --Matthiaspaul (talk) 20:43, 23 August 2015 (UTC)


 * Yes a history section is needed. A related need is some explanation of the domain and authority behind this convention. That is who dictates that this convention should be used for interpreting arithmetic expressions, within what disciplines is this accepted, and when may we rely on something we read following it? (I.e. texts after year such-and-such, texts from what disciplines, etc.) --Ericjs (talk) 18:29, 12 May 2016 (UTC)

Question
Anyone knows what would be the result of the equation y = 2x-x^2+6 when x = -0.5? Would it be 4.75 or 5.25? Is the negate applied to the x^2 first or the order 2 applied to x first, then negate?

xieliwei 16:39, 25 June 2006 (UTC)


 * Substitution, in an ambiguous case, must be carried out within parentheses. Thus, replacing x with -0.5 yields 2(-0.5) - (-0.5)^2 + 6 = -1 - (0.25) +6 = 4.75. Rick Norwood 14:01, 25 June 2006 (UTC)

I have a question also, when did mathamatics start using the Order of Operations? Was it something that came with the computer age or did the likes of Sir Isaac Newton or Pythagoras use it?--SerialCoyote 19:29, 17 November 2006 (UTC)


 * Obviously it was not as recent as the computer age. Just look at math journals and books of the 19th century or journals and books that appeared in the first half of the 20th century.  How could they have done without such things??  It seems weird to think they could have. Michael Hardy (talk) 17:04, 5 July 2008 (UTC)


 * That is an excellent question, and a subject worth researching. This article would benefit greatly from such information. Rick Norwood 13:33, 4 December 2006 (UTC)


 * Mathematics started using the Order of Operations immediately when multiplication was understood as a concept. It is not a convention that is true because it is agreed upon. It is a theorem (although not usually stated as such) that falls out from the definition of multiplication as repeated addition and the axioms of arithmetic (in particular, the distributive law and the multiplicative identity). Ditto for exponents because they are defined as repeated multiplication.Guildwyn (talk) 14:46, 5 July 2008 (UTC)


 * Wow - talk about a lightbulb moment! How did I clock up half a century without figuring that out? Blindingly obvious when stated like that. Thanks Guildwyn (if you are still around). 90.152.127.38 (talk) 14:42, 22 February 2017 (UTC)

Unfortunately, what Guildwyn says is not correct. Historically, various orders of operations have been used, and even today, as this article points out, different computer programs use different orders of operations. Because multiplication is repeated addition, it is convenient to do multiplication first. But it is not a theorem. Rick Norwood (talk) 12:24, 23 February 2017 (UTC)
 * I agree with Rick. Moreover, the order of operations was not considered before the invention of algebraic notation by François Viète during 16th century, while multiplication was understood as a concept 2000 years before (Euclid was certainly not the first one to understand this concept). D.Lazard (talk) 13:23, 23 February 2017 (UTC)

Paragraphs 2 and 3
Paragraph 3 is really just a synopsis of par. 2: I think it should go. GeneCallahan (talk) 20:18, 29 May 2017 (UTC)
 * ✅ Agreed. --Bill Cherowitzo (talk) 22:10, 29 May 2017 (UTC)


 * I put it back. It is there to emphasize the history of the idea, that it is not some modern invention but goes back to the beginnings of algebraic notation,. Rick Norwood (talk) 11:41, 30 May 2017 (UTC)


 * I also agree that it's largely redundant; I've tried another version. --JBL (talk) 16:58, 30 May 2017 (UTC)


 * OK. Rick has a good point and Joel's rewrite was better, but I still had that feeling of redundancy. I've tried another version which essentially combines the two paragraphs and so removes that feeling without losing that historical perspective.--Bill Cherowitzo (talk) 17:17, 30 May 2017 (UTC)

Mnemonics Example
"For example, using any of the above rules in the order "addition first, subtraction afterward" would incorrectly evaluate the expression[6]

10 − 3 + 2.

The correct value is 9 (and not 5, as if the addition would be carried out first and the result used with the subtraction afterwards)."

... Anyone who knows the order of operations would know that it doesn't matter (at the very least, in this situation), whether addition or subtraction goes first. *cough* — Preceding unsigned comment added by 67.253.251.43 (talk) 21:51, 6 January 2018 (UTC)

No. If we add first, we add 3 plus 2 to get five, and then subtract 10 - 5 to get 5. If we do the problem correctly, we subtract first to get 10 - 3 = 7. Then add 2 to get 9, the correct answer. You may think that "everybody knows that", but my children were taught out of a textbook that taught them to do it the first, wrong, way, so it is a real problem. Rick Norwood (talk) 13:01, 7 January 2018 (UTC)

exponentiation series
I checked the first cited source because I was skeptical and I was right to be: The Bronstein has exponentiation follow the normal left-associativity; that is unless some part of the notation says otherwise of course. In the most common handwritten notation (i.e. avoiding caret, arrow and similar symbols using superscripts instead), size and position of the exponents are such parts of the notation! If a term consisting of a simple basis and superscript configuration becomes an exponent itself, the uppermost exponent is superscripted doubly, thus reducing its size and increasing its relative elevation in the text twice each. If such a term gets an exponent on top, that exponent's size and relative elevation only change once, allowing for an easy differentiation. Of course the article correctly mentions the ambiguity of related notations, but deviating from the near-general left-associativity should maybe not be claimed to be the standard way. If no one else wants to make a change to the article or object to my argumentation, I will try to adjust it soon. Ninjamin (talk) 14:08, 1 April 2019 (UTC)
 * Your prose is difficult to understand because you mix typographical conventions (not the subject of this article) with the order of operations, the subject of this article. Nevertheless, the article follows the standard conventions, which are $$a^{b^c}=a^{(b^c)}$$ and $$(a^b)^c = a^{(bc)}=a^{bc}.$$ The explanation of this convention is clear: parentheses are not needed in both cases. So nothing need to be changed in the article. D.Lazard (talk) 14:38, 1 April 2019 (UTC)
 * Sorry for being hard to understand, I'm not a native english speaker. I wanted to say that the article's claim that exponentiation was generally right-associative is contradicted by the cited page in the Bronstein, which explicitly states that it be left-associative or, to be more precise with explicit exponentiation symbols, that a^b^c=(a^b)^c. (And as I see it the associativity of superscripts used instead of the '^' symbol is not governed by some order of operations, but by the size and position of the symbols, but as this last part is my opinion only I won't add it to the article.) Ninjamin (talk) 18:13, 1 April 2019 (UTC)
 * If the article contradicts Bronstein (I cannot check, as I have not access to this books), this means that Bronstein chooses a convention which is not the standard. In all mathematics text books and research articles $$a^{b^c}=a^{(b^c)}$$ (for example, see Fermat number and Double exponential function). Wikipedia cannot change this, as this would make not understandable a lot of mathematical texts. The only thing that is reasonable to do is to begin the second paragraph by "However, some authors and some computer systems ...", and move there the reference to Bronstein. D.Lazard (talk) 20:31, 1 April 2019 (UTC)
 * I have as of yet in no maths book seen a general right-associativity of exponentiation in operator notation (like a^b), only ever in the superscript notation (like $$a^b$$), that's what I'm saying. It's not a general exponentiation associativity rule, it's only for the superscript notation of exponentiation. Ninjamin (talk) 22:13, 1 April 2019 (UTC)

The order of operations needs to depend on the operations themselves, not the symbol used, nor the typeface used, to denote the operations. Otherwise we would need a potentially unlimited number of rules for every symbol and every typeface. As an aside, one interesting sidelight to the order of exponentiation occurred in Robert A. Heinlein's science fiction novel "The Number of the Beast", in which he interpreted 6^6^6 associating right instead of associating left. Science fiction author Larry Niven sent him a letter, explaining that exponentiation was right associative. Heinlein sent out a letter apologizing for the mistake.Rick Norwood (talk) 22:32, 1 April 2019 (UTC)
 * It needs to depend on the operation only and not the symbol, you say. But really it doesn't, as the paragraph about multiplication shows: In many physics books, multiplication with omitted multiplication sign has priority over division while multiplication with explicit multiplication sign has the same priority (i.e. is carried out from left to right). Also it's absolutely normal that additional order is given to the operation by the notation without using brackets, for example when using a fraction bar, it matters if you write $$\frac{a}{b+c}$$ or $$\frac{a}{b}+c$$. A similar thing occurs with exponentiation: $${a^b}^c \ne a^{b^c}$$ with the math codes being {a^b}^c and a^{b^c}. The difference in the size and position of the c is no accident or programming mistake. Ninjamin (talk) 08:38, 2 April 2019 (UTC)
 * This discussion came up earlier, and I was chided for saying it was physics books that mainly did this. As I said above, the important point is to avoid ambiguity.  As for you comment about symbols other than brackets being used as symbols of grouping, that is already covered in the article.  That does not lead to ambiguity. Rick Norwood (talk) 10:52, 2 April 2019 (UTC)
 * I agree that ambiguity is to be avoided (the ambiguous notation i mentioned above is uncommon for exactly that reason and it should be), but the article should not falsely say "exponentiation is right-associative in mathematics", as later it already says there is no general agreement. Unless there are further objections, I'll specify that passage with only the common superscript notation being right-associative, not exponentiation in general. Also the part about associatiativity between negation and exponentiation doesn't belong under "serial exponentiation". Ninjamin (talk) 16:21, 2 April 2019 (UTC)

word "operator" in section "definition"
I tried to clear something up in the section "definition", but someone disagreed and reverted it. So to avoid continuing the disagreement in the article's edit history only, I should ask about it here first. Please answer if you disagree with any of this. The word "operator" has two notions, in functional analysis it can mean a function between vector spaces (mostly function spaces with a metric, this notion usually does not appear in other contexts) and generally it can mean a formal symbol used to denote a function (mostly not consisting of letters). "Operator" is not, however, used generally to just denote any function, unlike "operation". So then, I think it is relevant for this article whether the described order only applies to operator (formal function symbol) included in an expression, or also when a function is denoted in another way, like by changing typeface or using superscript. Ninjamin (talk) 14:43, 11 June 2019 (UTC)
 * In mathematics, the original meaning of "operator" is "symbol denoting a function". This is only later that it has been extended to symbols denoting operations or functions whose domain is not well specified. This is the case for integration (for defining integrable functions one must already known integral). This is also the case in physics (Schrödinger operator). So the use of "operator" in this article is correct, as referring to the original meaning. It is not useful, and even confusing to change the formulation, as most readers of this article are not aware of functional analysis, nor of formal considerations. D.Lazard (talk) 15:17, 11 June 2019 (UTC)
 * Yes, I agree, functional analysis does not need to be mentioned in the article, I only mentioned it here to make sure I didn't miss a third meaning of "operator". So it is only the symbol for an operation, not the operation itself. That's exactly what I was talking about. Sometimes you can express an operation without a special symbol, that is, without an operator. For example, in physics a vector might be written in boldface and you can represent the absolute value of that vector by using the normal typeface. Or, given two numbers m and n, the nth power of m is written by using superscript, again without an operator (both m and n considered to be variables). Especially in the second example the order of operation still applies. But the given definition in the article only talks about representing an operation with operators. Ninjamin (talk) 08:53, 12 June 2019 (UTC)
 * You assert implicitly that a symbol must be a glyph or a combination of glyphs. This is not what Symbol says (there is not any specific definition of a symbol in mathematics). The beginning of this article is . This wide definition clearly includes typographical effects, such as font change and use of superscripts, and even the lack of a sign for meaning multiplication.
 * However, I agree that the unique sentence of section "Definition" that contains "operator" is awful. I'll try to improve it. D.Lazard (talk) 10:11, 12 June 2019 (UTC)
 * I see your point, but is this how "operator" is used in mathematics? None of the many lists of math symbols that I found in wikipedia mention symbols other than glyphs.
 * Also, I checked most of the citations, actually I didnt't find "operator" as symbol for an operation used by mathematicians, only by math teachers, computer scientists etc. Do mathematicians actually use "operator" in this meaning at all? Ninjamin (talk) 12:25, 2 July 2019 (UTC)
 * See List of mathematical symbols for symbols made from several glyphs. In particular, the symbol for set builder notation denotes an operation and consists of at least three glyphs. See also Legendre symbol, Jacobi symbol, ... The second sentence of Division (mathematics) contains "symbol" and "operator" in the same sense as in this article. As this article is watched by several professional mathematicians, and this is in the second sentence of the article, it is easy to conclude that the formulation is convenient for them.
 * Also, for writing mathematics, specifically at starting level, it is necessary to use words that are not mathematically defined, which have to be understood with their common language meaning. It is thus rather common that   different authors give them slightly different meaning, and this does not mean that some are wrong. Here, this is the case of "operator" and "symbol". D.Lazard (talk) 13:46, 2 July 2019 (UTC)
 * I guess I made the insensible assumption that you could always separate mathematical language from non-mathematical language about mathematics. So in the "Division" article, "symbols used for the division operator" means "symbols used as division operator", not "symbols representing the division operator", and many of the operator lists could be expanded to include many more non-glyph operators or at least mention that glyph-free operators are not included. I might look into that. Also, I want to thank you for your answers and your patience with me. Ninjamin (talk) 21:53, 12 July 2019 (UTC)

CLCstudent's edit
CLCstudent has now twice changed:


 * 10 − 3 + 2.

The correct value is 9 (not 5, as would be the case if you added the 3 and the 2 before subtracting from the 10). to


 * 10 − 3 + 2.

The correct value is 9 (in this case, it does not matter whether you do addition or subtraction first).

I would imagine that CLCstudent is mathematically sophisticated enough to understand that 10 - 3 + 2 is read, by a mathematician, as the sum of positive ten, negative three and positive two, in which case it does not matter whether you do addition or "subtraction" (addition of the opposite) first. However, that is not the way it is taught in some grade school textbooks, using "My Dear Aunt Sally". The students have not, at this level, been taught negative numbers, only the binary operations of addition and subtraction. They are taught that because in "My Dear Aunt Sally" "Aunt" (addition) comes before "Sally" (subtraction) they should first add the three and the two to get five, and then subtract the five from the ten to get a final answer of five. This is, as I'm sure CLSstudent knows, wrong. Nevertheless, it is the way many American schoolchildren are taught. Rick Norwood (talk) 11:23, 6 August 2019 (UTC)


 * This has also been discussed Talk:Order_of_operations above. On the question of what the article should say here, I agree with Rick Norwood.  I think the current form of the parenthetical is better than the older one, which did not specify what "the addition" was. --JBL (talk) 12:11, 6 August 2019 (UTC)