Talk:Order of operations/Archive 3

Archive 1

Archive 2

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Archive 4

Mnemonics and multiplication/division

The following material was recently added to the Mnemonics section and then reverted:

The order suggested by PEMDAS would incorrectly evaluate the expression

6 ÷ 2 * 3.

The correct value is 9 (and not 1, as if the multiplication would be carried out before the division). The 3 is not actually part of the denominator. Partially for this reason, the ISO_80000-2 standard for mathematical notation recommends using the solidus or fraction bar instead of the obelus symbol.

What follows is a discussion that took place on my talk page between me and the editor who proposed that addition. I'm copying it here so that we may further discuss it here on the talk page rather than on my user talk page. Mr. Swordfish (talk) 16:41, 16 January 2018 (UTC)

Re: the contribution I tried to make (that you removed) at https://wiki.riteme.site/w/index.php?title=Order_of_operations&oldid=818310968, was it unnecessarily redundant, too verbose, or do you think it needed a source reference in order to be credible or useful? Pseudothink (talk) 12:17, 5 January 2018 (UTC)

Hi, Pseudothink,

The main issue is that it needs a source. A side issue is that while it's fairly well established that A - B + C is almost always interpreted as A + -B + C, the corresponding application to multiplication/division is not so clear cut and may be ambiguous. At least that's my understanding; which is why we need to cite reliable sources - "my understanding" isn't good enough.

There seem to be thousands of threads on social media based upon something similar to your multiplication example (6 ÷ 2 * 3) with some contributors insisting on a strict interpretation of PEMDAS and others extrapolating from the convention for addition/subtraction. My take (again, "my take" is not good enough for Wikipedia) is that such an expression is ambiguous and a strict interpretation of PEMDAS may lead to an answer at variance with what many computer languages and calculators give.

In sum, I think a more nuanced version of what you had would pass muster; the key is to find reliable sources that treat the topic and accurately represent what the reliable sources say. If they disagree, then reflect that fact in the article. I think the article would be substantially improved by treating the multiplication/division example, we just need to be careful about how we treat it. Mr. Swordfish (talk) 16:24, 8 January 2018 (UTC)

A quick Google search turned up this: https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html As someone with an advanced degree in mathematics, I wholeheartely agree with it. Unfortunately, it's self-published and therefore not admissible as a reliable source. Let's see if we can turn up a better cite. Mr. Swordfish (talk) 17:23, 8 January 2018 (UTC)

Thanks, Mr. Swordfish, that was a very helpful response! Though I've used (and even financially supported) Wikipedia for quite some time, I'm still relatively new to contributing to it. My Computer Science background gives me a particular appreciation for the importance of following standards and best practices in this context, though. I need to self-educate a bit more to get my Wiki-fu up to snuff.

I'd based my previous contribution on feedback from my roommate, a math professor & PhD, and on a similar item I came up with from searching Google. I'll see if I can find a reliable source and improve the original contribution. I again just saw a similar example (6 ÷ 2 * (1 + 2)) pop up in a viral social media post, with most people arriving at the wrong answer, even some of whom I know to be college-educated STEM degree holders.

Unfortunately, this particular example is complicated (probably by design) by the use of the obelus symbol (÷), which is (per my previous update) discouraged from use in standard mathematical notation. Thus, despite being a popular viral problem, it may be a poor selection for use here, since it combines the issue of order of operations with that of correctly interpreting nonstandard mathematical notation. I'll see what my mathy roommate suggests. Thanks again! Pseudothink (talk) 02:44, 16 January 2018 (UTC)

Hi Pseudothink, and welcome to the talk page.

As indicated above, I think that it would improve the article to treat the multiplication example, we just need to be careful about how we do it. A core principle of Wikipedia is maintaining a neutral point of view. In particular, labeling something as "wrong" should only be done when the reliable sources are very clear about what is "right" and what is "wrong". I haven't seen that here.

My take is that expressions like (6 ÷ 2 * (1 + 2)) are ambiguous, and that various plausible interpretations of it are neither right nor wrong. I don't have a cite to back that up so I'm not going to put that opinion into the article. There's no harm in discussing it with your roommate, but anything in the article needs to have sources to verify it. Mr. Swordfish (talk) 17:35, 16 January 2018 (UTC)

It should be noted that the viral problem is not (6 ÷ 2 * (1 + 2)), it's (6 ÷ 2 (1 + 2)). It's a subtle, but important difference that is discussed in the body of the article. Implicit multiplication is treated differently to explicit multiplication by some calculators and professional style guides. This was the position of the American mathematical society in 2001, but I can't find a more recent reference. [1] StuartH (talk) 08:13, 22 February 2019 (UTC)

This subject has been discussed here for years, and is no closer to being resolved that the question of whether the even integers are a ring. Some authoritative sources say one thing, other equally authoritative sources say another, and there is no Bull Goose mathematician with the power to settle the question. The best we can do is note the difficulty and advise writers to avoid ambiguity.

This is really not a PEDMAS question, since all mathematicians agree that 6-2+3 is 7 and not 1 All authorities agree that PEDMAS is wrong, and should not still be taught in grade school. (It is widely taught, and the teachers who grew up on PEDMAS still teach children that the answer is 1, and are not about to change, because nobody can deny that A comes before S in PEDMAS. But there is nothing Wikipedia can do to fix that, except tell the truth and hope misled students discover it.)

The serious question is what 6/2*3 equals. Most mathematical sources say that the order of operations should not depend of the symbol used to denote that operation, and under that rule, 6/2*3=9, because ⁠6/2⁠*3=9. But there are many major physicists who have always done it the other way, and are not about to change just because mathematicians tell them to. Again, all Wikipedia can do is report the two views. Rick Norwood (talk) 16:09, 22 February 2019 (UTC)

It's certainly ambiguous, and the article should present whatever the sources say, but there's a surprising lack of sources when you look in to it. The AMS certainly used to take the "1" position, and frustratingly it currently gives 1/2(a+b) as an example of acceptable notation without actually clarifying its meaning. But the context suggests it should be 1/(2(a+b)), as is generally understood in both maths and physics literature (you would write it (a+b)/2 if you intend otherwise). StuartH (talk) 06:51, 24 February 2019 (UTC)

I can't imagine the math literature understanding 1/2(a+b) as different from ⁠1/2⁠(a+b), but I have gotten to the point where nothing surprises me. All we can do is report that people do things differently and that ambiguity should be avoided. Rick Norwood (talk) 12:57, 24 February 2019 (UTC)

Rick Norwood, you have been repeating the wrong idea that mathematicians interpret 1/2x as x/2 for years and years on this talk page without a single scrap of evidence to support it. It is wrong, it has always been wrong, and everything StuartH has said here is 100% correct. —JBL (talk) 14:22, 24 February 2019 (UTC)

I didn't intend to reopen a debate about what is right, I was just clarifying a subtle distinction here in the talk page. I think the article itself seems to have it right in presenting what the reliable sources say, but math literature that supports the (1/2)x side would be welcome if available. StuartH (talk) 10:16, 26 February 2019 (UTC)

Rick Norwood, I checked the Mathcentre source you added for your interpretation of 1/2x as (1/2)x. Nowhere in this source is this case covered. In fact, neither the slash operator nor multiplication by juxtaposition are used at all in this source. In particular, it is the order of the juxtaposition operation that must be explained here. I also checked the other reference (Bronstein) given for this interpretation, and I cannot find any example of this interpretation being used in the pages cited nor anywhere else I've searched within this text. However, in doing said search I did come across at least one counterexample. On page 277, the expression (1/ab) is used as part of an integral, where it is clear from context that it is taken to mean 1/(ab). This would seem to contradict the assertion that "you can't imagine the math literature understanding" such an interpretation. Also, from my personal academic experience in mathematics, physics and computer science in the US, I can state that I have never until now seen anyone interpret this type of expression in the way you do. It certainly may be ambiguous and should probably be avoided in some contexts, but I have seen this form used a number of times in my experience and on every occasion it was clear, without objection from anyone in the audience, that it meant precisely what you've said you can't imagine it meaning.Sethhoyt (talk) 15:18, 17 January 2020 (UTC)

And, over in rings, I've been repeating that some authors consider the even integers a ring. The people who disagree with that are just as certain that I am wrong. There is a strong tendency to be absolutely sure the first things we learn are absolutely true, and anyone who disagrees is absolutely wrong. All I'm saying is some people (and some calculators) do it one way and some do it the other way, and we should avoid ambiguity. Rick Norwood (talk) 15:09, 24 February 2019 (UTC)

The example of even integers is absolutely not comparable. It must be compared rather to the question whether 1 is prime. Both have been solved by choosing the convention that makes mathematics simpler. The difference is the date where the question arose: Rings have been popularized less than a century ago, and conventions take time for reaching a consensus.

The question of 1/2x should be rather compared to the question of choosing between $\sin x$ and $\sin(x):$ both are simply a question of notation and cannot be implied in any statement, contrarily to above two examples. The solution is very simple: where a doubt is possible, do not use the ambiguous notation, or state explicitly how it should be interpreted. IMO, any source that expresses things differently is either not reliable or presents a non-neutral point of view, and thus should be rejected.

Strongly related is the esthetic question. If $\sin x$ is often preferred to $\sin(x),$ this is for esthetic reasons (personally, like many other mathematicians, I never use $\sin x$ outside Wikipedia). Similarly the choice between ${\frac {a+b}{2}},{\frac {1}{2}}(a+b),(a+b)/2,(1/2)(a+b)$ depends on the size of the formula appearing in place of $a+b$ and whether this is a displayed or inline formula ($1/2(a+b)$ must be avoided). Note that Maple (software) returns ${\frac {a}{2}}+{\frac {b}{2}},$ whichever is the input, and interprets $/2$ as $2^{-1}.$ --D.Lazard (talk) 17:01, 24 February 2019 (UTC)

all I'm saying is The reason that I have repeatedly made a big deal of your wrongness on this point is that you do not only say that 1/2x is ambiguous and in some situations might be interpreted in a different way from what was intended. (That is true and uncontroversial.) Instead, you go on to make specific (false) claims about how certain ambiguous expressions (notably, 1/2x) are likely to be disambiguated in particular contexts (e.g., by professional mathematicians or in the mathematics literature). That is what I object to. --JBL (talk) 17:22, 24 February 2019 (UTC)

PEMDAS Paradox

I recommend including the information from the article with link below.

plus.maths.org/content/pemdas-paradox

68.96.208.77 (talk) 18:02, 22 April 2020 (UTC) Constructive Feedback

This is already in the article, and has been for years. It has also been discussed extensively on this Talk page. Many people on both sides are absolutely sure that they are right and the other side is wrong. This article cannot decide the question, only point out the problem and suggest avoiding any notation that will be widely misunderstood. Rick Norwood (talk) 11:48, 23 April 2020 (UTC)

The referenced article is pretty good -- it takes as its jumping-off point the silly $6\div 2(1+2)$ question, but it going beyond that. It's not crazy to check if there are uncited or weakly cited statements in the article that could be sourced to it. --JBL (talk) 15:13, 17 May 2020 (UTC)

We can always use more references. Rick Norwood (talk) 20:47, 17 May 2020 (UTC)

Division, subtraction, etc.

Moved from User talk:Joel B. Lewis § Order of operations.

 – JBL (talk) 12:37, 3 August 2020 (UTC)

First I don't edit wiki much but the order of operations is already a confusing topic for many learners. This article is littered with inaccuracy. Primary due to the multiple usages of negative examples. For the Mnemonic section, there is an example of addition/subtraction. When working the order of operations there is 2 ways to go about it. Either work left to right or use Additive inverse and Mixed division and multiplication.

The to keep things simple if you think of Addition and Subtraction as the exact same thing and Division and multiplication as the exact same thing you don't have to get confused. The steps even put these on the same step for that reason.

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Mnemonics Mnemonics are often used to help students remember the rules, involving the first letters of words representing various operations. Different mnemonics are in use in different countries.[8][10][11]

In the United States, the acronym PEMDAS is common.[12] It stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.[12] PEMDAS is often expanded to the mnemonic "Please Excuse My Dear Aunt Sally".[7] Canada and New Zealand use BEDMAS, standing for Brackets, Exponents, Division/Multiplication, Addition/Subtraction.[12] Most common in the UK, Pakistan, India, Bangladesh and Australia[13] and some other English-speaking countries is BODMAS meaning either Brackets, Order, Division/Multiplication, Addition/Subtraction or Brackets, Of/Division/Multiplication, Addition/Subtraction.[d][14][15] Nigeria and some other West African countries also use BODMAS. Similarly in the UK, BIDMAS is also used, standing for Brackets, Indices, Division/Multiplication, Addition/Subtraction.

These mnemonics may be misleading when written this way.[7] For example, misinterpreting any of the above rules to mean "addition first, subtraction afterward" would incorrectly evaluate the expression[7]

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).

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There is no reason to have this part in bold it only confuses people by creating a negative example. If they want to put bad math in the article maybe make a section for "Common Mistakes" or "Common misconception".

The equation 10-3+2 is exactly the same as 10 + (-3) + 2 and if you do the math there is no difference left to right or right to left. The problem is people who remove the negative. You could also read the problem as 10 -1 x 3 + 2 and in this case you again have the number -3.

If you treat the problem as 10 - (3 + 2) using the distributive property the problem is now 10-3-2 this is a completely different equation. I would love to just remove this. It is caused by incorrect application of the commutative property and a misunderstanding of numbers in general..

Next...

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Serial division A similar ambiguity exists in the case of serial division, for example, the expression 10 ÷ 5 ÷ 2 can either be interpreted as[citation needed]

10 ÷ ( 5 ÷ 2 ) = 4 or as

( 10 ÷ 5 ) ÷ 2 = 1 The left-to-right operation convention would resolve the ambiguity in favor of the last expression. Further, the mathematical habit of combining factors and representing division as multiplication by a reciprocal both greatly reduce the frequency of ambiguous division.

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This is just wrong again... Both of these sections should be removed to prevent continued misunderstanding or spreading misinformation on how to calculate using the Order of operation. I moved them to Common Mistakes as a compromise to fully removing them. But explaining these mistakes is to much and I have little formatting knowledge. They should just be removed but I will not keep fighting over this and repeat editing. You can adjust the new section if you think it could be formatted better or worded better if we are not going to agree on removing negative examples I hope we can agree on moving them to their own section. — Preceding unsigned comment added by 50.39.184.65 (talk) 02:53, 3 August 2020 (UTC)

I don't have time to discuss this right now, beyond to tell you that you are confused (albeit by a subtle point): there is no "correct answer" to "10 ÷ 5 ÷ 2" for several reasons; one is because it's not a question, but the more important one is that the operation ÷ is not associative and the section of the article correctly discusses the role of the left-to-right convention in disambiguating this ambiguous expression. Hopefully others can weigh in as well. --JBL (talk) 12:37, 3 August 2020 (UTC)

JBL Certainly, your version is better than the major revision that you reverted, and my attempt to improve that major revision, which you also reverted. And you are also correct in pointing out that division is not associative. You're too busy to work on this, and I have no objection to your most recent edit. However, I do feel like it is reasonable to interpret 10 ÷ 5 ÷ 2 as 10 x 1/5 x 1/2, by analogy to interpreting 10 - 5 - 2 as 10 + (-5) + (-2). This is, however, not a question that Wikipedia can solve. Rick Norwood (talk) 15:55, 3 August 2020 (UTC)

10 − 3 + 2 = 5 is just bad math, if you want to add the 2 and 3 do it right, move the +10 to the end and add -3 and 2 you get -1 and 10 its still the same 10-3+2 = -3+2+10 = (-3+2)+10 = -3+(2+10) it doesn't mater what order you do it in as long as you read the numbers correctly. The associative property may not apply to subtractions/division but that's only if you don't treat subtractions as negative numbers and treating division as multiplying by a fraction suddenly the associative property of multiplication and addition applies. Its not magic its math, there are rules that you have to adhere to. These examples of how to do math wrong are harmful not helpful.

$a-b+c=-b+c+a=(-b+c)+a=-b+(c+a)\neq a-(b+c)$ — Preceding unsigned comment added by 50.39.184.65 (talk) 00:31, 4 August 2020 (UTC)

$a-b+c=-b+c+a=(-b+c)+a=-b+(c+a)=a-(b-c)\neq a-(b+c)$ — Preceding unsigned comment added by 50.39.184.65 (talk) 00:36, 4 August 2020 (UTC)

So, based on the current version, it seems that "serial division" is not a special case... As said above, saying that 10 ÷ 5 ÷ 2 equals 4 is bad math, just as saying that 10 − 3 + 2 equals 5 is bad math. Also, I have yet to encounter a calculator or program that won't answer to 10/5/2 with a 1. Typing a serial division is a huge time saver, but most people still choose to type 10/(5*2) because they are unsure of the rules--Wikipedia should provide facts on this topic and not confused debate over a non-existent issue. Unless proof can be shown that there are cases out there (programs, languages, guidelines, peer reviewed publications) where a serial division is evaluated right-to-left (hence my original [citation needed]) I propose the Serial Division section to be removed from the special cases. Please note that Serial Exponentiation instead is a special case, because despite a commonly accepted rule it has been proven that there are exceptions (at the very least programs) that the reader should be aware of. This is not true for Serial Division. User:Danroa (talk) 15:53, 6 August 2020 (UTC)

I am in complete agreement with User:Danroa, I think I understand what the original poster was trying to warn users about but they have done in in a way that is not cautionary but almost presents it as a correct alternative solution. This can be extremely harmful and give teaches a head ache when they try to correct their students. I don't think it is completely incorrect to warn people that while using the order of operations it is left to right in almost every circumstances, they may have been hinting at the "Mixed division and multiplication" issues? I moved the division serialization to the same area of mnemonics and I removed the numbers and potential answers. Giving an incorrect answer to a math equation seems wrong in so many ways. It still explains some aspects of the order of operations that may be unclear to new users of it and I believe placing it with the mnemonics section is accurate as it is will the portion about the subtraction that has the same type of common mistake. Possible putting it in a section of "Left to Right" might help? Both of these pertain to the same issue of people being unsure what to do when an equation is on the same step of the order of operations (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction or Brackets, Exponents, Division/Multiplication, Addition/Subtraction ). We have the associative and commutative property as exceptions of calculating left to right, it really seems like a null point to argue over. I've always been a fan of converting my division into fractions and my subtractions in to signed numbers its easier to work with. — Preceding unsigned comment added by 50.39.184.65 (talk) 20:17, 8 August 2020 (UTC)

Does size of exponents matter?

I reverted a recent edit that states that the size of exponents matter. It does not state in what way the size matters, or how one decides when size matters. Thus, it is vague and uninformative and I reverted it. The poster restored it, claiming sources. Sadly, there are many conflicting rules about the order of operations, and standard books disagree. But even if some books say "size matters" that information is useless unless the rule they give for how size matters is stated. In any case, most books say top down unless parentheses, not different size exponents, say otherwise. Also, since a^(b^c) is often written without parentheses, has no simpler form, and almost always means the c power of the b power of a, introducing another rule were a^b^c sometimes means (a^b)^c can only cause confusion, especially since that rule doesn't say whether precedence is given to the exponent written larger or to the exponent written smaller (not to mention the problem of "how much larger or smaller before you use this rule). There is no reason anyone would ever write (a^b)^c in this way, since (a^b)^c can be simplified, using laws of exponents, to a ^(bc).

There are people who feel very strongly that the simple rule: exponents and roots first, multiplication and division second, addition and subtraction last, should have many exceptions in situations depending on the symbols used to express an operation and now on the size of the numbers involved. The inevitably results is confusion, and there is no advantage at all in confusion. In most cases, I think people who are taught a rule as students want that rule to be universal, even if it is clearly wrong. I can cite grade school textbooks that teach 3 - 2 + 1 = 0, and the teachers who teach that to children will not change their ways. They learned "My Dear Aunt Sally", add first, subtract second, and will continue to teach it, even though anyone who follows that rule after grade school will always get a wrong answer to that problem. (The answer is 2.)

I've been involved in a number of debates on this subject in Wikipedia, and nobody has ever changed their mind, not even about My Dear Aunt Sally, so I'm going to revert this business about large exponents and small exponents again, because it is non-standard and because it does not say what this supposed size rule is. Rick Norwood (talk) 12:03, 8 June 2021 (UTC)

I understand your point, though actually my edit says "if […] size and position of the exponents do not prescribe a specific order of operations" with the important word being "if". For some people size and position of the exponents might never matter, then my edit is still vacuously true. It just replaced the more specific rule with a generalized one that still includes it as a special case. Only if someone does indeed follow a rule of size and position of the exponents (whatever this rule may be), my edit comes into play and says that the top-down-rule does not overwrite it, which is obviously true, since that's exactly what "follow a rule of size and position of the exponents" means in this context.

Also as superscripts have many more uses than just to write powers and exponentials, the argument "you could just write a^bc" doesn't apply, sice many of them don't have this neat compatibility property. But these other uses are somewhat less common and so it might be ok to have to write parentheses everytime an ambiguity could arise.

In a formal context by the way, the rule does have exceptions, as a^b+cd≠a^(b+c)d shows, but in the given context one should assume a, b and c to be subexpressions of a^bc, so no argument here either.

The only question remaining is whether my edit was unnecessary, I think that's what you were aiming at in the first place. If noone actually does follow a deviating rule of size and position of the exponents, then my edit just causes confusion and is otherwise useless, even if true. So while size and position of exponents matter a lot in other cases, I guess regards to the associativity it is not necessary to mention them. Ninjamin (talk) 17:06, 8 June 2021 (UTC)

Implied Multiplication

I'd like to see a mention of implied multiplication and the controversy that surrounds it. As I understand it, there are two schools of thought.

The first is that implied multiplication implies brackets, so:

$ab=(a\times b)$

This means that:

$1\div ab=1\div (a\times b)={\frac {1}{ab}}$

The second school of thought is that implied multiplication does not imply brackets, so:

$ab=a\times b$

This means that (because of left to right operator precedence):

$1\div ab=1\div a\times b=(1\div a)\times b={\frac {b}{a}}$

As of now I don't feel qualified to write this as I don't know which bodies support which point of view. If someone feels that this is a worthy addition and is capable of giving it the context it requires then I would love to see this added. If there is a consensus that this would be a useful addition but nobody feels able to provide that context then I am happy to research and write. This may take some time, though.

St3f (talk) 06:25, 5 March 2020 (UTC)

This is discussed in section "Mixed division and multiplication". D.Lazard (talk) 09:28, 5 March 2020 (UTC)

I can't imagine anyone would object to improving the sourcing in that section (or to the article more broadly). --JBL (talk) 10:36, 5 March 2020 (UTC)

I can't believe I missed the "Mixed division and multiplication" section. That may not be as I would have chose to write it but it is the content that I was asking to be added. Thank you for bringing it to my attention. St3f (talk) 16:09, 5 March 2020 (UTC)

Also, by the way, while "left to right operator precedence" is taught in every American grade school, there is no such thing. It is a case of teachers teaching what they were taught, and paying no attention to the commutative and associative laws of addition and multiplication. To give one example, the easy way to add 3 + 8 + 7 is not left to right, but rather to add three and seven to get ten and then add the 8 to get 18. Rick Norwood (talk) 12:08, 5 March 2020 (UTC)

I like left-to-right operator precedence. I think that it makes mathematics more accessible. I think that we need a rule to ensure that people don't evaluate $1-2+3$ as $1-(2+3)$. I can either explain that subtraction is the addition of negative numbers and that the minus sign is bound rightly to the 2, making it $1+(-2)+3$ or I can explain that we evaluate left-to-right. Either works, but if I'm explaining this as simply as possible, I know which I'd choose (that's not to say that I might no use this with someone that is comforatble with algebra and arithmetic but it's not where I'd start). St3f (talk) 16:09, 5 March 2020 (UTC)

The problem with that is that, at least in the US, children are taught subtraction long before they are taught negative numbers, and so a child would be taught to say "you can't subtract two from one" but could work the problem if they added the three and the one to get four and then subtracted the two. Certainly it is very important to define the words "term" and "factor", but very few students in America are taught that. Which makes it difficult to explain that a term whose sign is negative can be treated as addition of the opposite term. Most American children are also not taught the difference between "opposite" and "reciprocal". When means that by the time I get them in college, I have to first get them to unlearn all the wrong stuff they were taught. My own children were taught My Dear Aunt Sally, and that the answer to 1-2+3 was "you can't take 2 from 1", while the answer to 10 - 2 + 3 was 5, because Aunt (addition) comes before Sally (subtraction). Rick Norwood (talk) 12:05, 6 March 2020 (UTC)

In the current paragraph, slashes are mentioned like they have anything to do with the ambiguity of implied multiplications, whereas whatever ambiguity exists with 1/2x also exists with 1÷2x --Bourriquet 42 (talk) 23:57, 6 August 2020 (UTC)

I agree, but not everyone agrees. Many think we need a different rule depending on which symbol is used to denote the operation. Bad idea, in my opinion. But Wikipedia cannot decide what mathematicians do. only observe and report. Rick Norwood (talk) 11:19, 7 August 2020 (UTC)

The "Mixed division and multiplication" section talks about the precedence of implicit multiplication over division, then gives an example that doesn't seem to relate to it at all. The example talks about precedence of all multiplication over division. I think there are two issues being conflated here. Acofokay (talk) 10:25, 19 February 2022 (UTC)

Look again. All of the examples are examples of implied multiplication. None uses any multiplication sign. On the other hand, there is a reference (without an example) that suggests that all multiplication should have precedence over division. Rick Norwood (talk) 16:07, 19 February 2022 (UTC)

Following up on the above, I doubt that even the most devout physicist would argue that 12 ÷ 3 × 2 was anything other than 8, which most physicists w2ould argue (because what you learned as an undergraduate seems like a law of nature) that 12 /3x was, if x = 2, certainly equal to 2. While a mathematician would argue that they symbol used for an operation should not change the result of that operation, so 12 /3x = 12 / 3 × 2 = 12 × 1/3 × 2 = 12 × 2/3 = 8. The moral of this story is to use parentheses to avoid any possibility of being misunderstood: 12/(3x) or 12/3 x, which are different. Rick Norwood (talk) 18:16, 19 February 2022 (UTC)

Functions.

What is the order of operations when a function is involved? Most examples are probably easy and obvious, however the example in front of me is "ln(3x - 4)^2." I assume the square is calculated before the natural logarithm; can someone confirm with a reference? Michael Hodgson (talk) 10:55, 5 March 2022 (UTC)]

Good question. The answer is implicitly given in functional notation. The rule of thumb is that when an ambiguity may occur, parentheses must be added. In particular, omitting parentheses after a function name (such as in "ln x") is allowed only for very simple arguments. So, your interpretation is wrong, as it supposes that the functional notation is used without parentheses. In other words, "ln(anything)" must be considered as an indecomposable entity, and, in your example, this is the whole entity that is squared. Practically, when a opening parenthesis can be interpreted as a part of a functional notation, the function argument(s) is (are) delimited by the corresponding closed parenthesis. Similarly, $\ln(3x-4)\,y=y\ln(3x-4),$ and $\ln(3x-4)\,y\neq \ln(3xy-4y).$ Note also that $\ln ^{2}(3x-4)=\ln \ln(3x-4)=(\ln \circ \ln )(3x-4)=\ln(\ln(3x-4)).$

This should be clarified here and in functional notation, but I do not a good source for that. Nevertheless, I am pretty sure that some exist. D.Lazard (talk) 12:03, 5 March 2022 (UTC)

Thank you. Michael Hodgson (talk) 02:52, 6 March 2022 (UTC)

Must children be taught to perform operations from left to right

An editor insists that there is a rule of mathematics and gives it as addition and subtraction must be performed from left to right, and as multiplication and division must be performed from left to right. There is no such rule. The actual rule is that subtraction is addition of the opposite and that division is multiplication by the reciprocal. In other words 3 - 2 + 1 = 2 not because the operations are performed from left to right, but because subtraction is addition of the opposite. Consider, for example 2 - 3 + 1. Any mathematician will get 0. But following the grade school rule a student will say, you can't take 3 from 2, so the operation is undefined. There are many similar examples. 57 + 75 - 57 is trivially equal to 75. To force children to add 57 and 75 and then to subtract 57 is much harder and a waste of time. 35 + 81 + 65 is easy to do if you add the 35 and 65 first to get 100, and then add the 81 to get 181, harder worked from left to right.

I teach calculus, and most of my students struggle with elementary arithmetic because they have been taught so many rules that are either unnecessary or wrong in grade school, that I have to teach them arithmetic at the same time I teach them calculus. But people who learned these rules in grade school often insist the grade school rules are right, and cite grade school textbooks as references.

I would appreciate comments on this question. Rick Norwood (talk) 15:02, 14 November 2021 (UTC)

I think it would be fairer to say that there are two equivalent ways to correctly interpret 3 - 2 + 1. One is that the additions and subtractions should happen from left to right, so it equals (3 - 2) + 1 and not 3 - (2 + 1). Another is to say that the minus turns into adding a negative, and then the three terms in 3 + -2 + 1 can be added in any order. As you say, the flexibility that comes with this more advanced way of thinking about it is very pragmatically useful for evaluating an expression. Note that the article currently discusses this way of thinking about it in the middle of the Defintiion section, but it might help for this to be elaborated on (including an example like your 57 + 75 - 57 to show why it's useful) and contrasted with the more basic way of thinking about it. Maybe this topic should get its own section. But beginners need to understand that 3 - 2 × 1 + 1 means (3 - (2 × 1)) + 1, and not 3 - ((2 × 1) + 1) or (3 - 2) × (1 + 1), before they can even begin to understand your point about reordering the terms. This is why the left to right rule is always part of the explanation of the mnemonic. Danstronger (talk) 16:11, 14 November 2021 (UTC)

It is correct that A - B = A + (-B), but the symbol "-" on the left hand side is clearly a binary operator which *is* left-associative, and it is important that kid's know that they can't simplify right away when they spot " ... 5 - 5 ... " in some expression (namely, in case the first 5 is also preceded by a "-"). I think it is much safer, and also much simpler, and therefore better, to explain to kids that all these operations (- and /) must be done left to right. This is definitely not a complicated rule, but a simple and natural prescription/recipe that makes life easier. They can understand later that they can mix the order if they rewrite A - B as A + (-B), and A/B as A · B^-1. — MFH:Talk 14:30, 1 July 2022 (UTC)

"left associate" misleading?

In the article it's written that some operators are "maybe misleadingly called left-associative". I wonder how that could be misleading (and why this correct terminology should be avoided as the author seems to imply with this remark). Apart from the fact that this is the precise, correct term to name this situation, "to associate" clearly means "to group together", so "left associative" obviously means to group together the left side, what else could it mean? — MFH:Talk 14:38, 1 July 2022 (UTC)

I agree, and will make the change accordingly.Rick Norwood (talk) 10:46, 2 July 2022 (UTC)

undone "Order of Operations by emphasizing concepts" section

A recent editor, MFH, asked for comments on his explanation of the order of operations. I'll just make two.

"Do the most important things first - in life and in mathematics. Repeaters cause Bigger Change - they are more important."

The problem with this is that which is "most important" is ambiguous. For example, some people might consider multiplication "more important" than roots.

1. Perform the operations “just like you read a sentence.” ("Left to Right" implies something new.)

This is not a rule, though many teachers tell students it is. For example, to work 5 x 73 x 2 it is much easier to do (using the commutative and associative laws) 5 x 2 = 10, 10 x 73 =730. Students should not be told that something is a "rule" when it isn't.

Rick Norwood (talk) 11:08, 14 August 2022 (UTC)

My username is gregwelch8 and I am the one that posted this edit. Here is the link to the edit under discussion - section name: “Order of Operations by emphasizing concepts”:

[2]

---

Thank you for the comments.

Mathematics has lots of ‘rules’ that have exceptions.  

Replace rule with guidelines if it is an issue…or, teachers can state up front that there are exceptions.

The goal at subject intro is to focus on teaching the concept.

To the degree possible, we want to connect concepts to what students already know -

and to what they need to know.  This method does that.

1. You wrote, “The problem with this is that which is "most important" is ambiguous. For example, some people might consider multiplication "more important" than roots.”

The message of that ‘rule’ is that Bigger Change is more important - in life and in math. Notice that Bigger Change is capitalized and in bold so as to emphasize those words. Bigger Change is easy to visualize on a number line.  Show a couple of egs starting at zero.  Multiple a number by 5 and compare it to adding 5 to a number.  Yes, there are exceptions - like 0, 1, fractions…but students in 3rd/4th grade have done many more problems in which multiplication causes a Bigger Change than addition. This rule connects to what the student already knows so its easy to remember/easy to visualize.

Without this rule/generality, how should students remember which operation goes first?  

2. Regarding calculating “just like you read a sentence.”  This connects ‘reading’ a math problem to reading a sentence.  There is nothing new to remember (like, “left to right”).

Performing operations (on the same level) - “just like you read a sentence” - allows a student to solve these problems and get the correct answer.  You brought up a helpful exception that makes the problem easier to solve.  

Without this rule, how will the student proceed?

3. In addition to the above, please comment of the the simplicity of pairing operations with a simple word that both describes them and groups them: Repeaters and Singles.  These simple words simply Order of Operations, working with Fact Families, remembering operations by pairs and more.

The edit I proposed ended with this:  In summary: by pairing operations, delaying exponents until introduced, and separating the parens topic: we give students 2 choices rather than 6; we change the Q&A from answers like, “because the M comes before the A” to concepts fundamental to math.

Hoping that elementary school teachers and tutors will share their views on this method.

Using PEMDAS, BODMAS or any acronym to teach Order of Operations detracts from time that could have been used to reinforce some of the keys of math. Explanations that follow PEMDAS introduction such as, "because the M comes before the A" or discussing which conjunction (and/or) connects which operation results is hours spent discussing an acronym rather than reinforcing some of the fundamentals of mathematics.

The link posted above also has three graphics that help explain this method.  Please read the entire entry to better understand this method. GregWelch8 (talk) 20:12, 14 August 2022 (UTC)

I completely agree with user:Rick Norwood's removal of this material. It's confusing, poorly written, it contains WP:MOS violations, and it contains unencyclopedic language. This article is about order of operations. It's not a place for teachers or pedagogical experts to expound on their favourite technique for teaching the order of basic operations to lower level students. Meters (talk) 08:27, 15 August 2022 (UTC)

The section I submitted is written for the students..and teachers, tutors, parents that help these “lower level” students.  There are millions of hours per year wasted on teaching by acronym - so this is an important topic/section. The section that immediately follows, “Special Situation” seems like the place to focus on the higher level issues - including the exceptions.

Teaching Order of Operations by concepts (rather than acronym) is so much simpler.  Let’s go thru an example.  Look at the example (graphic) I posted.  Explain how you would solve if you were in elementary school.

My entry:

1.Mark the Repeaters and calculate - just like you read a sentence.

2.Only Singles remain.  Again, just like you read a sentence.”

What’s more, using this method enables students to reinforce some of the fundamentals of math rather than discuss letters in an acronym. Also, PEMDAS and BODMAS have too many choices - some of which students have no knowledge of (exponents).

Specifically, why does this method confuse you?

The grammar/style problems you cite. I will review the Wiki style guide again and then look for errors.  If you want to share any/make some edits - please do so.  (Rick Norwood just informed me that using capital letters for emphasis is against Wiki standards. I will fix.)

And…what do you think about the terms Repeaters and Singles?  Pairing the operations is a crucial step for multiple reason - including Order of Operation. Don’t you think it is peculiar that add/subt and mult/div are pairs - yet we have no name for the pairs?  If they have a Common Core they have a common (descriptive) characteristic. This is it….as a generality.

I look forward to your reply. GregWelch8 (talk) 11:15, 15 August 2022 (UTC)

Agree with Rick Norwood and Meters above. Wikipedia is not the place to advocate for a particular pedagogy, especially one that appears to be novel. See WP:NOTHOWTO and WP:NOR. Mr. Swordfish (talk) 13:35, 15 August 2022 (UTC)

I'll just mention one additional point. Using capital letters for emphasis is against the standards of Wikipedia.

Thousands of people have ideas about the order of operations. It is not a subject for beginners to try to figure out for themselves. The rules are standard and simple and the attempts at explanations of the rules often make understanding worse. Rick Norwood (talk) 09:48, 15 August 2022 (UTC)

Rick,

Thank you for the tip on the use of capital letters.  I will fix the edit and review the Wiki style guide.

You state the rules and standards are simple and also wrote (on another talk page) that the various acronyms are confusing and useless. So, for a third time, please share these rules you prefer so that we can compare methods.  

You mention that there are thousands of methods/ideas about how to do Order of Operations - please share the best one. Perhaps we should start with it?  See how we can improve. That is what I hoped would be done with my submission.

This is not that complex.  Fewer, simpler steps, connecting/integrating with what is known and what needs to be learned is better.  

--- ps

Again, please comment on the terms Repeaters and Singles?  I just sent the following to Meters and would appreciate your comment:

Just pairing the operations is a crucial step for multiple reasons - including Order of Operations. Don’t you think it is peculiar that add/subt and mult/div are pairs - yet we have no name for the pairs?  If they have a Common Core they have a common characteristic. GregWelch8 (talk) 11:50, 15 August 2022 (UTC)

My opinion on the terms Repeaters and Singles is irrelevant to what we do here as editors. Even if I thought the terms were the greatest thing since Arabic numerals, unless there is a reliable published source supporting it we can't put it in the article. From the help pages:

Wikipedia does not publish original thought. All material in Wikipedia must be attributable to a reliable, published source. Articles may not contain any new analysis or synthesis of published material that serves to reach or imply a conclusion not clearly stated by the sources themselves.

Perhaps you could point to some published source to support the material you added. Then we'd have something to discuss. Cheers. Mr. Swordfish (talk) 13:42, 15 August 2022 (UTC)

Mr. Swordfish,

As Wiki editors, one would assume that we are experts on the issue under review and are allowed to edit/change words. As editors of an encyclopedia we are also expected to summarize primary and secondary sources.  Lastly, “Routine calculations do not count as original research, provided there is consensus among editors...”

I believe all the information in the section I submitted is well accepted, easily understood, and not in dispute.

Eg, this is well accepted: Operations are in the order of the order of magnitude of their effect. I.e. exponents cause a bigger change than multiplication, which cause a bigger change than addition. Do you dispute this?  

Parenthesis are on top to allow you to override the order when required.  Do you dispute this?  

Two specific words you did bring up: Repeaters and Singles.

There are many diagrams that show add/subt and mult/div on the same level.  They just did not apply a descriptive, well understood one-word name.

This author did group and name the operations; she also points out the obvious benefits.  This one of many articles critical of PEMDAS.  

https://leafandstemlearning.com/2015/07/why-i-will-never-use-pemdas-to-teach-html/

“Using the term “multiplicative operations” not only reinforces new standards and vocabulary requirements that students be able to differentiate between additive and multiplicative operations, it also brings both multiplication and division into a single step.  Students stop thinking about multiplication and division as two separate concepts, but as one concept.”

For the 3^rd and 4^th graders learning Order of Operations, I think Repeaters/Singles accomplishes the same thing.  It’s shorter, better understood and more descriptive.

Other than Repeaters and Singles, what statements/words are under dispute?

------

A general note about who the Wiki Order of Operations page is intended.  At present, the Wiki page on the subject is primarily a high level discussion that makes it great for math experts.  

It is not very useful for the main target audience. There millions of students per year that need to learn Order of Operations….and all we offer in our summary on the subject is PEMDAS…with the very editor that made the most recent edit/version (Rick Norwood) calling acronyms “confusing and useless.”  

So, our only contribution so far on the basics is “useless.” We still need a small section on the acronyms given they are still used, but we can do better.  We owe readers a method that teaches by well accepted concepts (rather than acronym) and that integrates what students know at the time of presentation.  This method also needs to integrate with what a student needs to know in the future.   GregWelch8 (talk) 16:58, 15 August 2022 (UTC)

The article as it stands, and has stood for some time, gives the rules by means of examples, in the introduction, and with a brief, correct definition, in the first section: "Definition". PEMDAS and other acronyms are given later in the article, because they are used by so many grade schools. I have never seen them used in college or in a college textbook or by anyone who uses mathematics professionally. They need to be in the article so the incorrect information that is still taught in grade schools can be corrected.

Attempts to change the language in which these rules are expressed is original research, and has no place in Wikipedia.

To give just one example of why this does not work, consider the statement above "Parenthesis are on top to allow you to override the order when required." This article is about the order of operations. Parentheses are not an operation, they are a symbol of grouping, discussed in its own article. The relationship between parentheses and the order of operations is already explained briefly and clearly here. There is no need to confuse the two. Rick Norwood (talk) 10:41, 16 August 2022 (UTC)

Rick,

There are millions of elementary math students and their educators - per year - that would benefit from useful information on Order of Operations.  At present, the Wiki page offers little of value to the main target audience of this subject.

These readers want to know what it is and how to do it.  Wikipedia is not a how-to manual but it should share and overview what’s important and useful about the subject at hand.

The “Description” section does a good job of explaining the Order of Operations.  The current overview of the how it is done in practice (“Mnemonics”) is a “useless” dated acronym.  We provide it as a “service to educators” who continue to use this confusing method.

A section titled, “Order of Operations emphasizing concept” will help millions of students with Order of Operations as well as several other concepts in math.

As far as I know (and have asked several times in the discussion above), there is nothing about the content of my edit that is under dispute.  Only that my first version was too educational, some words needed to be changed from bold to plain text, and most recently: it does not belong on Wikipedia because its new information/primary research.

-----

We have already Paired the operations.  There are hundreds of graphics showing the operations in Pairs.  Google “Order of Operations” and select images.  You will see pyramids, stacked blocks, charts and drawings that group the operations by Pairs.  

Lauren elegantly summarizes the value of naming the Pairs in this article:

https://leafandstemlearning.com/2015/07/why-i-will-never-use-pemdas-to-teach-html/

She points out that the Pair Names: reinforce new standards and vocabulary requirements; they bring both multiplication and division into a single step; and students stop thinking about multiplication and division as two separate concepts, but as one concept.

Naming the pairs is not just about simplifying Order of Operations.  It will also help with Balancing Equations, Fact Families, Fractions and more.  I can provide explicit egs for those not involved in elementary education.

Is the value of giving the basic Pairs a name under contention?

You also indicated that I was attempting to change the language. “Repeated addition” is commonly used as a term to describe multiplication.   This article/online text book from Elementary Math at EDC uses the term “Repeated Addition” in describing multiplication 10 times:

https://elementarymath.edc.org/resources/multiplication/

This video from Khan Academy uses "repeated addition" in the titled, "Multiplication as repeated addition."  Mr. Khan then uses the term “Repeated Addition” 5 times in the presentation.

https://www.khanacademy.org/math/cc-third-grade-math/intro-to-multiplication/imp-multiplication-intro/v/multiplication-as-repeated-addition

It is not much of a stretch to call Repeated Addition (multiplication) a Repeater.  This is an something we do as editors/experts in the field under discussion.

The current Pair-name choices are Repeaters/Singles and Multiplicative/Additive, or both. Repeaters/Singles is shorter, more descriptive, and a term that is used and well understood by elementary students and educators.  (Also, Multiplicative could be interpreted as referring to multiplication.  Same issue with Additive.)

If another name comes along that widely understood, more descriptive, etc, we will change the entry. In the mean time, we owe the main audience of this section some help.

-----

Here’s a draft of the revised edit.  This could be placed under the “Mnemonics” section.  Hoping we can edit/improve on this and post. I would imagine the exact words we use will be under dispute/under discussion. Which is good.

Order of Operations emphasizing concepts

Using PEMDAS, BODMAS or any acronym to teach Order of Operations detracts from time that could have been used to reinforce some of the keys of math. The Order of Operations is the order of the magnitude of their effect. That is, exponents/roots cause Bigger Change than Repeaters, which cause Bigger Change than Singles.

The example to the right illustrates the simplicity to providing two choices rather than the six provided by PEMDAS.  When expos/roots are introduced later, they cause even Bigger Change.  The first step will change to mark expos/roots and calculate.

Parentheses (parens) are not operations but symbols to indicate Special Treatment.

In math, that special treatment is:  Do First.  

Link to image used in the example (will not post the caption):

https://commons.wikimedia.org/w/index.php?title=File:OrderOps_eg.jpg&direction=prev&oldid=681884876#/media/File:OrderOps_eg.jpg

----- ps

The “Definition” section also needs an edit.  In the first paragraph, the Pairs need a name:

multiplication and division  (repeaters)

addition and subtraction (singles) GregWelch8 (talk) 18:02, 17 August 2022 (UTC)

What you're trying to add here is simply counter to what Wikipedia does - Wikipedia explicitly is not a place to introduce new ideas. See WP:NOR (really, please read it) for details. This is why the content you are trying to add is being disputed. I would suggest that you contact some publishers of mathematics textbooks and see if you can get them interested directly. After we see it in some published books Wikipedia might adopt it, but not before. MrOllie (talk) 18:23, 17 August 2022 (UTC)

MrOlie,

The ideas are not new. The post that you replied to is full of references to the current use of the term Repeater and to works which group the basic operations pairs. It also includes an article reference to the benefits of naming the operations in Pairs. Did you see these? Please comment on them.

The benefits of Pair names are not in dispute. We do not need to numerous references to verify facts, simple concepts or the value of separating a complex issue into individual concepts. We do not need to wait years for textbook revisions/curriculum changes in order to share factual information by topic that helps our readers.

Put aside your "new idea" objection for one minute. RickNoland, Meters, MrSwordish...same request.

Answer:

1. Is labeling the basic operations in pairs helpful (or extremely helpful) for several topics in elementary math?
2. Is anything in the edit proposed under dispute/non-factual?

If the answers are "extremely" and "nothing", why are we not sharing?

Because there is a debate about how many references/publications we need to publish facts/improvements?

How far behind would you like Wikipedia to be? GregWelch8 (talk) 19:21, 17 August 2022 (UTC)

You made it a change. It was undone. Four editors have tried to explain why your changes have been undone. No-one has supported your changes. We edit by consensus, and you do not have it. I suggest that you drop this. This is getting into WP:BLUDGEONING territory. Meters (talk) 19:36, 17 August 2022 (UTC)

I read the links you provided, and while they use variations of the word 'repeat', they do not actually support the content you are attempting to add. Re your statement that we do not need references - yes we do. See WP:V. This is one of Wikipedia's core policies. Many things are helpful and yet do not belong on Wikipedia, for example most things listed in Wikipedia:What Wikipedia is not. To answer your last question: I would like Wikipedia to be exactly as far behind as the reliable sources we can cite are. - MrOllie (talk) 19:36, 17 August 2022 (UTC)

MrOllie,

Thank you for reviewing the links.  The links show the group pairs, they use the repeat word/concept, and name the pairs.  So we will just disagree on this.  

Not sure where I wrote that references are not needed.  The amount of supporting material/references required for known facts is much less than posting on political issues, etc.  

I posted references and asked if there was any dispute about the value of pairing the basic operations?  That is, what needs to be verified?  Share that and I will find more references.

Again, I think we are simply down to how many references are needed to support.  No one is objecting to the value or factual content of the edit.

It is a shame that we are ignoring the beauty of math.  The simplified method means something in math.  It means it is the answer.

-------

Meters,

I read the objections and questions that are posted and and reply specifically to them. The current objection is, “it’s new information/ideas.”  The long post I made today addressed this objection with several links documenting the use of the terminology and the grouping of operations as pairs.  

I believe we down to the issue many references are required for known facts/issues that are undisputed and which will greatly benefit readers.

When we are no longer exchanging ideas, I will wait several days, then I file a RfC or Dispute (if I can confirm one or both of them reaches a broader group).  Hopefully, some participants have children or grandchildren struggling with Order of Ops/elementary math.  

If the audience is not diverse, I will send letters/links summarizing this exchange to some generalists at Wikipedia that believe in providing helpful, factual information…and not waiting years for paper textbook updates. GregWelch8 (talk) 21:03, 17 August 2022 (UTC)

@Rick Norwood@Meters@Mr swordfish@MrOllie

How about a link near the top of the current Order of Operations page to a basic discussion of the subject?  Only editors interested in elementary math need view.  

Could start it with the current/obligatory section describing Order with an acronym.  Then a section on understanding it by concepts.  See how it develops.   GregWelch8 (talk) 20:37, 3 September 2022 (UTC)

See WP:NOTHOWTO. This is simply not what Wikipedia is for. MrOllie (talk) 20:39, 3 September 2022 (UTC)

a compendium that contains information on all branches of knowledge GregWelch8 (talk) 20:55, 3 September 2022 (UTC)

Wikipedia is not an indiscriminate collection of information. - MrOllie (talk) 20:57, 3 September 2022 (UTC)

What I said last time: You made it a change. It was undone. Four editors have tried to explain why your changes have been undone. No-one has supported your changes. We edit by consensus, and you do not have it. I suggest that you drop this. This is getting into WP:BLUDGEONING territory. Time to WP:DROPTHESTICK. Meters (talk) 03:23, 4 September 2022 (UTC)

BODMAS' O

The O in bodmas is by no means universally held to be Orders. In fact, this is the first time that I have heard it to be specifically called this. Others that I have heard include:

Over (Similar to divide or brackets) Other (would include exponents) Of (Similar to divide) Order (Close to orders)

Essentially, the O is there really just to add a much needed vowel in the middle of a group of consonents. This is especially true when considering that it is really only useful for children - as an acronym wouldn't do for all of the mathematical operators (really). MATH HAS A HATER... ME!

When I was taught the Bomdas Rule (Ireland) 'O' meant "Of means multiply, and must be done as if inside brackets" for questions such as 2 + 1/2 of 4. Regards, MartinRe 10:39, 7 May 2999 (UTC)

Ref for above [3] MartinRe 10:41, 7 May 2006 (UTC)

I was also taught this (in England), and I believe it is wrong. Most people would evaluate "1/2 of 3 + 7" as 5, some as 8.5, and none as 7.16666 (as required by the "Of" version). Maproom (talk) 09:35, 4 April 2008 (UTC)

In the school text books of the school where I went in the North-east of England, about six years ago, it almost always said the O stood for 'of'.--Jcvamp 06:08, 14 February 2007 (UTC)

• BODMAS Should be included as it is used in England as the mnemonic used —Preceding unsigned comment added by 219.79.73.236 (talk) 02:51, 15 February 2010 (UTC)

Order of multiple vertical division

The article currently explains a ÷ b ÷ c, and/or a/b/c as left associative. But it doesn’t explain the order of vertical division like this:

${\frac {\left({\frac {a}{b}}\right)}{c}}\neq {\frac {a}{\left({\frac {b}{c}}\right)}}\qquad or:{\frac {\frac {a}{b}}{c}}\neq {\frac {a}{\frac {b}{c}}}$

I believe the consensus is top-to-bottom. But looking at a math video like this (see YouTube vid ‘6SzZ_jAHasE’ by MindYourDecisions), and the comments, that’s apparently not well known.

One of the arguments there was that ‘wider’ bars show precedence. Using standard MathMl here shows exactly that, which suggests that the above is a division of one rational by another. However, ignoring that rendering effect (and sure if that’s a “math rule” at all), the question remains: if there’s no explicit disambiguation, what’s the order here? My vote is top-to-bottom.

If there is no consensus, or an authoritative source in this, we could leave it out. Otherwise, I hope somebody who knows how to write math symbols could add it here, or on the page on division (neither mentions it). Abel (talk) 19:54, 17 September 2022 (UTC)

What you are describing is a complex fraction, and it is dealt with at fraction. Fractions do have some minor distinctions from division, so it is properly dealt with on that article and not on this one. MrOllie (talk) 20:50, 17 September 2022 (UTC)

Thanks @mrollie. I search ‘division’, didn’t think to look there. I see they write there:

If, in a complex fraction, there is no unique way to tell which fraction lines takes precedence, then this expression is improperly formed, because of ambiguity. So 5/10/20/40 is not a valid mathematical expression, because of multiple possible interpretations, e.g. as

which seems to contradict this article. Or at the very least it is confusing. Also, isn’t a compound fraction in that form just the same as a sequence of divisions? I agree with conclusion there, but I do feel like it has a place in this article as well: it’s another way of writing divisions. Abel (talk) 00:43, 18 September 2022 (UTC)

No, a fraction is usually a form of notation and indicates a particular number (1/2 is a number, just as .5 is a number). It is not an 'operation' as discussed on this article. You can often simplify a fraction with division, but sometimes the fraction is the endpoint. MrOllie (talk) 00:46, 18 September 2022 (UTC)

Thanks again for clarifying. Here in The Netherlands we have one of those basic math rules you learn as a child “divide by a fraction is the same as multiplying the inverse (of that fraction)”. We show such divisions by using the horizontal line (under it there being the fraction, sometimes over it as well), to make this rule visibly clearer.

We also learn that the original division operator “÷” looks like it does because it shows the horizontal line with dots as placeholders, meaning that 14÷6 is equal to $14 \over 6$.

Finally, we learn not to use the “/“ for division if we can help it, as it can be ambiguous. Using the horizontal line (as with fractions) and then simplifying and crossing out like terms, is (here at least) the preferred method. I’ve also seen this method on quite a few websites and on Wikipedia math/science articles probably for the clarity it gives.

I.e. you rarely see 3x + 12x/5 + y = 3, but you’d see: $3x+{12x \over 5}+y=3$.

But if that’s not the common, or agreed upon way in math to do division (at least internationally like in published papers or for the purposes of this article for what we mean with “operations”) then that’s it. I’m not a specialist on the matter, and I just take my information from several people being confused over the order of operations, and they equal (just as in the fractions article you referred to) division by “/“ with division by “—”. Abel (talk) 21:04, 18 September 2022 (UTC)

Note to D. Lazard

"Order in which mathematical operations are performed" is clear. "Order to perform mathematical operations" would be read by most people as "Give an order to a person that they must perform a mathematical operation", an entirely different meaning of the word "order".Rick Norwood (talk) 22:07, 26 October 2022 (UTC)

To editor Rick Norwood: I have changed the SD to "Performing order of mathematical operations". THis has 43 characters, which is acceptable for WP:SD40. I hope that it is not confusing. D.Lazard (talk) 22:32, 26 October 2022 (UTC)

Thanks. That is much clearer. Rick Norwood (talk) 10:22, 27 October 2022 (UTC)

Mathematics Bodmas

3-3×6+2=? 168.167.26.138 (talk) 19:45, 16 November 2022 (UTC)

I don't think anyone here wants to deprive you of the pleasure of reading this wonderfully written article. Dhrm77 (talk) 20:08, 16 November 2022 (UTC)