I wrote a (very long) blog post about those viral math problems and am looking for feedback, especially from people who are not convinced that the problem is ambiguous.
It’s about a 30min read so thank you in advance if you really take the time to read it, but I think it’s worth it if you joined such discussions in the past, but I’m probably biased because I wrote it :)
Hi! Nice blog post. Since you asked for feedback I’ll point out the one thing I didn’t really understand. You explain the difference between the calculators by showing excerpts from the manuals and you highlight that in the first manual, implicit multiplication is prioritised. But the text you underlined only refers to implicit multiplication involving special expressions(?) like pi, e, sqrt or log, and nothing about “regular” implicit multiplication like 2(1+3). So while your photos of the calculator results are great proof that the two models use a different order of operations, to me the manuals were a bit confusing since they did not actually seem to prove your point for the example math problems you are discussing. Or maybe I missed something?
only refers to implicit multiplication involving special expressions(?) like pi, e, sqrt or log, and nothing about “regular” implicit multiplication like 2(1+3)
That was a very astute observation you made there! The fact is, for the very reason you stated, there is in fact no such thing as “implicit multiplication” - it is a term which has been made up by people who have forgotten Terms (the first thing you mentioned) and The Distributive Law (the second thing you mentioned). As you’ve noted., these are 2 different rules, and lumping them together as one brings exactly the disastrous results you might expect from lumping different 2 rules together as one…
See here for explanation of all the various rules, including textbook references and proofs.
I would do the mighty parentheses first, and then the 2 that dares to touch the mighty parentheses, finally getting to the run-of-the-mill division. Hence the answer is One.
What the heck are you all fighting about? It’s BODMAS.
They’re arguing about whether Distribution is Multiplication or not. Spoiler alert: it isn’t, it’s Brackets.
I’d would be great if you find the time to read the post and let me know afterwards what you think. It actually looks trivial as a problem but the situation really isn’t, that’s why the article is so long.
It actually looks trivial as a problem
Because it actually is.
that’s why the article is so long
The article was really long because there were so many stawmen in it. Had you checked a Maths textbook or asked a Maths teacher it could’ve been really short, but you never did either.
I was being facetious. I will try to find the time to read the post, but I know already that the problem isn’t trivial. It involves, above all else, human comprehension, which is a very iffy thing, to say the least.
I’ve seen a calculator interpret 1 ÷ 2π as ½π which was kinda funny
An e-calculator I’m guessing? (either that or Texas Instruments) Desmos USED TO interpret that correctly, but then they made a change with automatically turning division into fractions and broke it (because if you’ve specified division then it’s not a fraction) dotnet.social/@SmartmanApps/111164851485070719
I believe it was a app , yes
All calculators that are listed in the article as following weak juxtaposition would interpreted it that way.
And they’re all wrong dotnet.social/@SmartmanApps/111164851485070719
What if the real answer is the friends we made along the way?
That’d be good, but what I’ve found so far here is a whole bunch of people who don’t like being told the actual facts of the matter! 😂
FACT CHECK 4/5
a solidus (/) shall not be followed by a multiplication sign or a division sign on the same line
There’s absolutely nothing wrong with doing that. The order of operations rules have everything covered. Anything which follows an operator is a separate term. Anything which has a fraction bar or brackets is a single term
most typical programming languages don’t allow omitting the multiplication operator
Because they don’t come with order of operations built-in - the programmer has to implement it (which is why so many e-calculators are wrong)
“.NET IDE0048 – Add parentheses for clarity”
Microsoft has 3 different software packages which get order of operations wrong in 3 different ways, so I wouldn’t be using them as an example! There are multiple rules of Maths they don’t obey (like always rounding up 0.5)
Let’s say we want to clean up and simplify the following statement … o×s×c×(α+β) … by removing the explicit multiplication sign and order the factors alphabetically: cos(α+β) Nobody in their right mind would remove the explicit multiplication sign in this case
This is wrong in so many ways!
- you did multiplication before brackets, which violates order of operations rules! You didn’t give enough information to solve the brackets - i.e. you left it ambiguous - you can’t just go “oh well, I’ll just do multiplication then”. No, if you can’t solve Brackets then you can’t solve ANYTHING - that is the whole point of the order of oeprations rules. You MUST do brackets FIRST.
- the term (α+β) doesn’t have a coefficient, so you can’t just randomly decide to give it one. It is a separate term from the rest Is there supposed to be more to this question? Have you made this deliberately ambiguous for example?
- if the question is just to simplify, then no simplification is possible. You’ve not given any values to substitute for the pronumerals
- (α+β) is presumably (you’ve left this ambiguous by not defining them) a couple of angles, and if so, why isn’t the brackets preceded by a trig function?
- As it’s written, it just looks like a straight-forward multiplying and adding pronumerals except you didn’t give us any values for the pronumerals meaning no simplfication is possible
- if this was meant to be a trig question (again, you’ve left out any information that would indicate this, making it ambiguous) then you wouldn’t use c, o, or s for your pronumerals - you’ve got a whole alphabet left you can use. Appropriate choice of pronumerals is something we teach in Maths. e.g. C for cats, D for dogs. You haven’t defined what ANY of these pronumerals are, leaving it ambiguous
Nobody will interpret cos(α+β) as a multiplication of four factors
- as originally written it’s 4 terms, not 1 term. i.e. it’s not cos(α+β), it’s actually oxsxxx(α+β), since that can’t be simplified. And yes, that’s 4 terms multiplied!
From those 7 points, we can see this is not a real Maths problem. You deliberately made it ambiguous (didn’t say what any of the pronumerals are) so you could say “Look! Maths is ambiguous!”. In other words, this is a strawman. If you really think Maths is ambiguous, then why didn’t you use a real Maths example to show that? Spoiler alert: #MathsIsNeverAmbiguous hence why you don’t have a real example to illustrate ambiguity
Implicit multiplications of variables with expressions in parentheses can easily be misinterpreted as functions
No they can’t. See previous points. If there is a function, then you have to define what it is. e.g. f(x)=x². If no function has been defined, then f is the pronumeral f of the factorised term f(x), not a function. And also, if there was a function defined, you wouldn’t use f as a pronumeral as well! You have the whole rest of the alphabet left to use. See my point about we teach appropriate choice of pronumerals
So, ambiguity really hides everywhere
No, it really doesn’t. You just literally made up some examples which go against the rules of Maths then claimed “Look! Maths is ambiguous!”. No, it isn’t - the rules of Maths make sure it’s never ambiguous
IMHO it would be smarter to only allow the calculation if the input is unambiguous.
Which is exactly what calculators do! If you type in something invalid (say you were missing a bracket), it would say “syntax error” or something similar
force the user to write explicit multiplications
Are you saying they shouldn’t be allowed to enter factorised terms? If so, why?
force notation that is never ambiguous
We already do
but that would lead to a very convoluted mess that’s hard to read and write
In what way is 6/2(1+2) either convoluted or hard to read? It’s a term divided by a factorised term - simple
providing context that makes it unambiguous
In other words, follow the rules of Maths.
Links about various potentially ambiguous math notations
Spoiler alert: they’re not
“Most ambiguous phrases and notations in maths”
e.g. fx=f(x), which I already addressed. It’s either been defined as a function or as pronumerals, therefore nothing ambiguous
“Absolute value notation is ambiguous”
No, it’s not. |a|b|c| is the absolute value of a, times b, times the absolute value of c… which you would just write as b|ac|. Unlike brackets you can’t have nested absolute values, so the absolute value of (a times the absolute value of b times c) would make no sense, especially since it’s the EXACT same answer as |abc| anyway!
In-line power towers like
Left associativity. i.e. an exponent is associated with the term to its left - solve exponents right to left
People saying “I don’t know how to interpret this” doesn’t mean it’s ambiguous, nor that it isn’t defined. It just means, you know, they need to look it up (or ask a Maths teacher)! If someone says “I don’t know what the word ‘cat’ means”, you don’t suddenly start running around saying “The word ‘cat’ is ambiguous! The word ‘cat’ is ambiguous!” - you just tell them to look it up in a dictionary. In the case of Maths, you look it up in a Maths textbook
Because the actual math is easy almost everybody has an opinion on it
…and any of them which contradict any of the rules of Maths are demonstrably wrong
Most people also don’t know that with weak and strong juxtaposition there are two conflicting conventions available
…and Maths teachers know that both of them are made-up and not real things in Maths
But those mnemonics cover just the basics. The actual real world is way more complicated and messier than “BODMAS”
Nope. The mnemonics plus left to right covers everything you need to know about it
Even people who know about implicit multiplication by juxtaposition dismiss a lot of details
…because it’s not a real thing
Probably because of confirmation bias and/or because they don’t want to invest so much time into thinking about stupid social media posts
…or because they’re a high school Maths teacher and know all the rules of Maths
the actual problem with the ambiguity can’t be explained in a quick comment
Yes it can…
Forgotten rules of Maths - The Distributive Law (e.g. a(b+c)=(ab+ac)) applies to all bracketed Terms, and Terms are separated by operators and joined by grouping symbols
Bam! Done! Explained in a quick comment
FACT CHECK 5/5
most people just dismiss that, because they “already know” the answer
Maths teachers already know how to do Maths. Huh, who would’ve thought? Next thing you’ll be telling me is English teachers know the rules of grammar and how to spell!
and a two-sentence comment can’t convince them how and why it’s ambiguous
Literally NOTHING can convince a Maths teacher it’s ambiguous - Maths teachers already know all the rules of Maths, and which ones you’re breaking
Why read something if you have nothing to learn about the topic that’s so simple that you know for a fact that you are right
To fact check it for the benefit of others
At this point I hope you understand how and why the original problem is ambiguous
At this point I hope you understand why it isn’t ambiguous. Tip: next time check some Maths textbooks or ask a Maths teacher
that one of the two shouldn’t even be a thing
Neither of them is a thing
not everybody shares your opinion and preferences
Facts you mean. The rules of Maths are facts
There is no mathematically true
There absolutely is! You just chose not to ask any experts about it
the most important thing with this “viral math” expressions is to recognize that
…they are all solvable by following the rules of Maths
One could argue that there should also be a strong connection between coefficients and variables (like in r=C/2π)
There is - The Distributive Law and Terms
it’s fine to stick to “BIDMAS” in school but be aware that that’s not the full story
No, BIDMAS and left to right is the full story
If you encounter such discussions in the wild you could just post a link to this page
No, post a link to this order of operations thread index - it has textbook references, proofs, memes, worked examples, the works!
Hi, I’m stupid, is it 1+2 first, then multiple it by 2, then divide 6 by 6?
Or is it 1+2, then divide 6 by 2, then multiple?
I think it’s the first one but I’ve got no idea.
It’s the first, as per The Distributive Law and Terms. It could only ever be the second if the 6/2 was in brackets. i.e. (6/2)(1+2).
It’s actually “both”. There are two conventions. One is a bit more popular in science and engineering and the other one in the general population. It’s actually even more complicated than that (thus the long blog post) but the most correct answer would be to point out that the implicit multiplication after the division is ambiguous. So it’s not really “solvable” in that form without context.
You’d think we would’ve solve this with Einstein or Aristotle or something.
Indeed it was already solved more than 100 years ago. The issue isn’t that it’s “ambiguous” - it isn’t - it’s that people have forgotten what they were taught (students don’t get this wrong - only adults). i.e. The Distributive Law and Terms.
Having read your article, I contend it should be:
P(arentheses)
E(xponents)
M(ultiplication)D(ivision)
A(ddition)S(ubtraction)
and strong juxtaposition should be thrown out the window.Why? Well, to be clear, I would prefer one of them die so we can get past this argument that pops up every few years so weak or strong doesn’t matter much to me, and I think weak juxtaposition is more easily taught and more easily supported by PEMDAS. I’m not saying it receives direct support, but rather the lack of instruction has us fall back on what we know as an overarching rule (multiplication and division are equal). Strong juxtaposition has an additional ruling to PEMDAS that specifies this specific case, whereas weak juxtaposition doesn’t need an additional ruling (and I would argue anyone who says otherwise isn’t logically extrapolating from the PEMDAS ruleset). I don’t think the sides are as equal as people pose.
To note, yes, PEMDAS is a teaching tool and yes there are obviously other ways of thinking of math. But do those matter? The mathematical system we currently use will work for any usecase it does currently regardless of the juxtaposition we pick, brackets/parentheses (as well as better ordering of operations when writing them down) can pick up any slack. Weak juxtaposition provides better benefits because it has less rules (and is thusly simpler).
But again, I really don’t care. Just let one die. Kill it, if you have to.
I think weak juxtaposition is more easily taught
Except it breaks the rules which already are taught.
the PEMDAS ruleset
But they’re not rules - it’s a mnemonic to help you remember the actual order of operations rules.
Just let one die. Kill it, if you have to
Juxtaposition - in either case - isn’t a rule to begin with (the 2 appropriate rules here are The Distributive Law and Terms), yet it refuses to die because of incorrect posts like this one (which fails to quote any Maths textbooks at all, which is because it’s not in any textbooks, which is because it’s wrong).
Except it breaks the rules which already are taught.
It isn’t, because the ‘currently taught rules’ are on a case-by-case basis and each teacher defines this area themselves. Strong juxtaposition isn’t already taught, and neither is weak juxtaposition. That’s the whole point of the argument.
But they’re not rules - it’s a mnemonic to help you remember the actual order of operations rules.
See this part of my comment: “To note, yes, PEMDAS is a teaching tool and yes there are obviously other ways of thinking of math. But do those matter? The mathematical system we currently use will work for any usecase it does currently regardless of the juxtaposition we pick, brackets/parentheses (as well as better ordering of operations when writing them down) can pick up any slack. Weak juxtaposition provides better benefits because it has less rules (and is thusly simpler).”
Juxtaposition - in either case - isn’t a rule to begin with (the 2 appropriate rules here are The Distributive Law and Terms), yet it refuses to die because of incorrect posts like this one (which fails to quote any Maths textbooks at all, which is because it’s not in any textbooks, which is because it’s wrong).
You’re claiming the post is wrong and saying it doesn’t have any textbook citation (which is erroneous in and of itself because textbooks are not the only valid source) but you yourself don’t put down a citation for your own claim so… citation needed.
In addition, this issue isn’t a mathematical one, but a grammatical one. It’s about how we write math, not how math is (and thus the rules you’re referring to such as the Distributive Law don’t apply, as they are mathematical rules and remain constant regardless of how we write math).
It isn’t, because the ‘currently taught rules’ are on a case-by-case basis and each teacher defines this area themselves
Nope. Teachers can decide how they teach. They cannot decide what they teach. The have to teach whatever is in the curriculum for their region.
Strong juxtaposition isn’t already taught, and neither is weak juxtaposition
That’s because neither of those is a rule of Maths. The Distributive Law and Terms are, and they are already taught (they are both forms of what you call “strong juxtaposition”, but note that they are 2 different rules, so you can’t cover them both with a single rule like “strong juxtaposition”. That’s where the people who say “implicit multiplication” are going astray - trying to cover 2 rules with one).
See this part of my comment… Weak juxtaposition provides better benefits because it has less rules (and is thusly simpler)
Yep, saw it, and weak juxtaposition would break the existing rules of Maths, such as The Distributive Law and Terms. (Re)learn the existing rules, that is the point of the argument.
citation needed
Well that part’s easy - I guess you missed the other links I posted. Order of operations thread index Text book references, proofs, the works.
this issue isn’t a mathematical one, but a grammatical one
Maths isn’t a language. It’s a group of notation and rules. It has syntax, not grammar. The equation in question has used all the correct notation, and so when solving it you have to follow all the relevant rules.
Nope. Teachers can decide how they teach. They cannot decide what they teach. The have to teach whatever is in the curriculum for their region.
Yes, teachers have certain things they need to teach. That doesn’t prohibit them from teaching additional material.
That’s because neither of those is a rule of Maths. The Distributive Law and Terms are, and they are already taught (they are both forms of what you call “strong juxtaposition”, but note that they are 2 different rules, so you can’t cover them both with a single rule like “strong juxtaposition”. That’s where the people who say “implicit multiplication” are going astray - trying to cover 2 rules with one).
Yep, saw it, and weak juxtaposition would break the existing rules of Maths, such as The Distributive Law and Terms. (Re)learn the existing rules, that is the point of the argument.
Well that part’s easy - I guess you missed the other links I posted. Order of operations thread index Text book references, proofs, the works.
You argue about sources and then cite yourself as a source with a single reference that isn’t you buried in the thread on the Distributive Law? That single reference doesn’t even really touch the topic. Your only evidence in the entire thread relevant to the discussion is self-sourced. Citation still needed.
Maths isn’t a language. It’s a group of notation and rules. It has syntax, not grammar. The equation in question has used all the correct notation, and so when solving it you have to follow all the relevant rules.
You can argue semantics all you like. I would put forth that since you want sources so much, according to Merriam-Webster, grammar’s definitions include “the principles or rules of an art, science, or technique”, of which I think the syntax of mathematics qualifies, as it is a set of rules and mathematics is a science.
That doesn’t prohibit them from teaching additional material
Correct, but it can’t be something which would contradict what they do have to teach, which is what “weak juxtaposition” would do.
a single reference
I see you didn’t read the whole thread then. Keep going if you want more. Literally every Year 7-8 Maths textbook says the same thing. I’ve quoted multiple textbooks (and haven’t even covered all the ones I own).
mathematics is a science
Actually you’ll find that assertion is hotly debated.
Correct, but it can’t be something which would contradict what they do have to teach, which is what “weak juxtaposition” would do.
Citation needed.
I see you didn’t read the whole thread then. Keep going if you want more. Literally every Year 7-8 Maths textbook says the same thing. I’ve quoted multiple textbooks (and haven’t even covered all the ones I own).
If I have to search your ‘source’ for the actual source you’re trying to reference, it’s a very poor source. This is the thread I searched. Your comments only reference ‘math textbooks’, not anything specific, outside of this link which you reference twice in separate comments but again, it’s not evidence for your side, or against it, or even relevant. It gets real close to almost talking about what we want, but it never gets there.
But fine, you reference ‘multiple textbooks’ so after a bit of searching I find the only other reference you’ve made. In the very same comment you yourself state “he says that Stokes PROPOSED that /b+c be interpreted as /(b+c). He says nothing further about it, however it’s certainly not the way we interpret it now”, which is kind of what we want. We’re talking about x/y(b+c) and whether that should be x/(yb+yc) or x/y * 1/(b+c). However, there’s just one little issue. Your last part of that statement is entirely self-supported, meaning you have an uncited refutation of the side you’re arguing against, which funnily enough you did cite.
Now, maybe that latter textbook citation I found has some supporting evidence for yourself somewhere, but an additional point is that when providing evidence and a source to support your argument you should probably make it easy to find the evidence you speak of. I’m certainly not going to spend a great amount of effort trying to disprove myself over an anonymous internet argument, and I believe I’ve already done my due diligence.
Citation needed.
So you think it’s ok to teach contradictory stuff to them in Maths? 🤣 Ok sure, fine, go ahead and find me a Maths textbook which has “weak juxtaposition” in it. I’ll wait.
Your comments only reference ‘math textbooks’, not anything specific
So you’re telling me you can’t see the Maths textbook screenshots/photo’s?
outside of this link which you reference twice in separate comments but again, it’s not evidence for your side, or against it, or even relevant
Lennes was complaining that literally no textbooks he mentioned were following “weak juxtaposition”, and you think that’s not relevant to establishing that no textbooks used “weak juxtaposition” 100 years ago?
We’re talking about x/y(b+c) and whether that should be x/(yb+yc) or x/y * 1/(b+c).
It’s in literally the first textbook screenshot, which if I’m understanding you right you can’t see? (see screenshot of the screenshot above)
you have an uncited refutation of the side you’re arguing against, which funnily enough you did cite.
Ah, no. Lennes was complaining about textbooks who were obeying Terms/The Distributive Law. His own letter shows us that they all (the ones he mentioned) were doing the same thing then that we do now. Plus my first (and later) screenshot(s).
Also it’s in Cajori, but I didn’t find it until later. I don’t remember what page it was, but it’s in Cajori and you have the reference for it there already.
you should probably make it easy to find the evidence you speak of
Well I’m not sure how you didn’t see all the screenshots. They’re hard to miss on my computer!
P.S. if you DID want to indicate “weak juxtaposition”, then you just put a multiplication symbol, and then yes it would be done as “M” in BEDMAS, because it’s no longer the coefficient of a bracketed term (to be solved as part of “B”), but a separate term.
6/2(1+2)=6/(2+4)=6/6=1
6/2x(1+2)=6/2x3=3x3=9
Starting a new comment thread (I gave up on reading all of them). I’m a high school Maths teacher/tutor. You can read my Mastodon thread about it at Order of operations thread index (I’m giving you the link to the thread index so you can just jump around whichever parts you want to read without having to read the whole thing). Includes Maths textbooks, historical references, proofs, memes, the works.
And for all the people quoting university people, this topic (order of operations) is not taught at university - it is taught in high school. Why would you listen to someone who doesn’t teach the topic? (have you not wondered why they never quote Maths textbooks?)
#DontForgetDistribution #MathsIsNeverAmbiguous
I’m curious if you actually read the whole (admittedly long) page linked in this post, or did you stop after realizing that it was saying something you found disagreeable?
I’m a high school Maths teacher/tutor
What will you tell your students if they show you two different models of calculator, from the same company, where the same sequence of buttons on each produces a different result than on the other, and the user manuals for each explain clearly why they’re doing what they are? “One of these calculators is just objectively wrong, trust me on this, #MathsIsNeverAmbiguous” ?
The truth is that there are many different math notations which often do lead to ambiguities.
In the case of the notation you’re dismissing in your (hilarious!) meme here, well, outside of anglophone high schools, people don’t often encounter the obelus notation for division at all except for as a button on calculators. And there its meaning is ambiguous (as clearly explained in OP’s link).
Check out some of the other things which the “÷” symbol can mean in math!
#MathNotationsAreOftenAmbiguous
did you stop after realizing that it was saying something you found disagreeable
I stopped when he said it was ambiguous (it’s not, as per the rules of Maths), then scanned the rest to see if there were any Maths textbook references, and there wasn’t (as expected). Just another wrong blog.
What will you tell your students if they show you two different models of calculator, from the same company
Has literally never happened. Texas Instruments is the only brand who continues to do it wrong (and it’s right there in their manual why) - all the other brands who were doing it wrong have reverted back to doing it correctly (there’s a Youtube video about this somewhere). I have a Sharp calculator (who have literally always done it correctly) and most of my students have Casio, so it’s never been an issue.
trust me on this
I don’t ask them to trust me - I’m a Maths teacher, I teach them the rules of Maths. From there they can see for themselves which calculators are wrong and why. Our job as teachers is for our students to eventually not need us anymore and work things out for themselves.
The truth is that there are many different math notations which often do lead to ambiguities
Not within any region there isn’t. e.g. European countries who use a comma instead of a decimal point. If you’re in one of those countries it’s a comma, if you’re not then it’s a decimal point.
people don’t often encounter the obelus notation for division at all
In Australia it’s the only thing we ever use, and from what I’ve seen also the U.K. (every U.K. textbook I’ve seen uses it).
Check out some of the other things which the “÷” symbol can mean in math!
Go back and read it again and you’ll see all of those examples are worded in the past tense, except for ISO, and all ISO has said is “don’t use it”, for reasons which haven’t been specified, and in any case everyone in a Maths-related position is clearly ignoring them anyway (as you would. I’ve seen them over-reach in Computer Science as well, where they also get ignored by people in the industry).
Has literally never happened. Texas Instruments is the only brand who continues to do it wrong […] all the other brands who were doing it wrong have reverted
Ok so you’re saying it never happened, but then in the very next sentence you acknowledge that you know it is happening with TI today, and then also admit you know that it did happen with some other brands in the past?
But, if you had read the linked post before writing numerous comments about it, you’d see that it documents that the ambiguity actually exists among both old and currently shipping models from TI, HP, Casio, and Canon, today, and that both behaviors are intentional and documented.
There is no bug; none of these calculators is “wrong”.
The truth is that there are many different math notations which often do lead to ambiguities
Not within any region there isn’t.
Ok, this is the funniest thing I’ve read so far today, but if this is what you are teaching high school students it is also rather sad because you are doing them a disservice by teaching them that there is no ambiguity where there actually is.
If OP’s blog post is too long for you (it is quite long) i recommend reading this one instead: The PEMDAS Paradox.
In Australia it’s the only thing we ever use, and from what I’ve seen also the U.K. (every U.K. textbook I’ve seen uses it).
By “we” do you mean high school teachers, or Australian society beyond high school? Because, I’m pretty sure the latter isn’t true, and I’m skeptical of the former. I thought generally the ÷ symbol mostly stops being used (except as a calculator button) even before high school, basically as soon as fractions are taught. Do you have textbooks where the fraction bar is used concurrently with the obelus (÷) division symbol?
Ok so you’re saying it never happened, but then in the very next sentence you acknowledge that you know it is happening with TI today
You asked me what I do if my students show me 2 different answers what do I tell them, and I told you that has never happened. None of my students have ever had one of the calculators which does it wrong.
that both behaviors are intentional and documented
Correct. I already noted earlier (maybe with someone else) that the TI calculator manual says that they obey the Primary School order of operations, which doesn’t work with High School order of operations. i.e. when the brackets have a coefficient. The TI calculator will give a correct answer for 6/(1+2) and 6/2x(1+2), but gives a wrong answer for 6/2(1+2), and it’s in their manual why. I saw one Youtuber who was showing the manual scroll right past it! It was right there on screen why it does it wrong and she just scrolled down from there without even looking at it!
none of these calculators is “wrong”.
Any calculator which fails to obey The Distributive Law is wrong. It is disobeying a rule of Maths.
there is no ambiguity where there actually is.
There actually isn’t. We use decimal points (not commas like some European countries), the obelus (not colon like some European countries), etc., so no, there is never any ambiguity. And the expression in question here follows those same notations (it has an obelus, not a colon), so still no ambiguity.
i recommend reading this one instead: The PEMDAS Paradox
Yes, I’ve read that one before. Makes the exact same mistakes. Claims it’s ambiguous while at the same time completely ignoring The Distributive Law and Terms. I’ll even point out a specific thing (of many) where they miss the point…
So the disagreement distills down to this: Does it feel like a(b) should always be interchangeable with axb? Or does it feel like a(b) should always be interchangeable with (ab)? You can’t say both.
ab=(axb) by definition. It’s in Cajori, it’s in today’s Maths textbooks. So a(b) isn’t interchangeable with axb, it’s only interchangeable with (axb) (or (ab) or ab). That’s one of the most common mistakes I see. You can’t remove brackets if there’s still more than 1 term left inside, but many people do and end up with a wrong answer.
By “we” do you mean high school teachers, or Australian society beyond high school?
I said “In Australia” (not in Australian high school), so I mean all of Australia.
Because, I’m pretty sure the latter isn’t true
Definitely is. I have never seen anyone here ever use a colon to mean divide. It’s only ever used for a ratio.
Do you have textbooks where the fraction bar is used concurrently with the obelus (÷) division symbol?
All my textbooks use both. Did you read my thread? If you use a fraction bar then that is a single term. If you use an obelus (or colon if you’re in a country which uses colon for division) then that is 2 terms. I covered all of that in my thread.
EDITED TO ADD: If you don’t use both then how do you write to divide by a fraction?
FACT CHECK 2/5
The behaviour is intended and even carefully documented in the manual
…and yet still a bug (I saw at least one other person point this out to you)
A few years ago, there was a Microsoft feature intended for people in China, but people who weren’t in China were getting that behaviour. i.e. a bug. It was documented and a deliberate design choice for people in China, but if you weren’t in China then it’s a bug. Just documenting a design choice doesn’t mean bugs don’t happen. A calculator giving a wrong answer is a bug
weak juxtaposition is only used by old calculators
Based on the comments in the above video, the opposite is true - this problem first arose in '96
because they are scientific calculators.
So the person programming it is far more likely to need to check their Maths first - bingo!
TI (Texas Instruments) also has some calculators that use strong juxtaposition and some products that use weak juxtaposition
…and some that use both! i.e. some follow Terms but not The Distributive Law. As I said to begin with, these are 2 DIFFERENT rules, and you can’t just lump them together as one
evaluate 1/2X as 1/(2X)
Which is correct, as per Terms
while other products may evaluate the same expression as 1/2X from left to right
What you mean is they evaluate it as 1/2xX, since 1/2X and 1/(2X) are the same thing
it would be necessary to group 2X in parentheses
No, not necessary, since 2a=(2xa) by definition, alluded to in Cajori in 1928…
Sharp is a bit of an exception here, because all their other scientific calculators seem to
…follow all the rules of Maths, always. There’s something to be said for making sure you’re doing it right. :-)
Google uses the same priority for explicit and implicit multiplication
…and they will actually remove brackets I have put in and replace them with their own (“hi” to all the people who say you can fix any calculator by “just add more brackets” - Google doesn’t CARE what brackets you’ve added!)
Desmos and GeoGebra try to force the user into using fractions (which is a good design decision if you ask me)
It’s not, because a ÷ isn’t a fraction bar. They’re joining 2 terms into one and thus sometimes changing the answer
A lot of other tools like programming languages, spreadsheets, etc. don’t allow implicit multiplication syntax at all
It’s not that they don’t allow it, it’s that it’s not provided with the language by default in the first place! Most languages only provide you with some numbers, operators, and a few functions (like round), and it’s up to the programmer to implement the rest. Welcome to why there are so many wrong e-calculators
let you choose if you want weak or strong juxtaposition
…which is a red flag to not use that calculator!
This gives you more control about how you like the calculator to behave in these situations
I’m not sure it does. I’d have to switch on “strong juxtaposition” (the only kind there is) and see what else has been disobeyed in Maths. e.g. Google removing my brackets and adding different ones
Wolfram|Alpha only uses strong juxtaposition between named variables, but weak juxtaposition for everything else. This might seem strange and inconsistent at first but is probably the least surprising behaviour for most people
I find any exceptions to following the rules of Maths surprising! No, you can’t just make up your own rules
many textbooks, “a/bc” is intended to denote a/(bc)
a/bc=a/(bc) in every textbook
Wolfram Language, it means (a/b)×c
Welcome to “we’re gonna add brackets to what you typed in and change the answer”
a multiplication sign has been omitted
…then that means it’s not “multiplication” - it’s Terms and/or The Distributive Law. The “M” in the mnemonics refers literally to multiplication signs, nothing else
Multiplication and division have the same priority, they are “mathematically speaking” the same operation. This also applies to addition and subtraction. One is just the inverse function of the other
Yep, and The Distributive Law and Factorising are the inverse of each other
no rule about “multiplication before division” or “division before multiplication” they always have the same priority
…and Brackets is always first, so in this case it doesn’t even matter
In no way do any of the mnemonics represent any standard or norm in mathematics
Yes they do - mnemonics represent the actual order of operations rules
most children don’t become mathematicians later in life and if they do, they will learn all the other important stuff about the order of operations later
No, they won’t. Year 8 is the last time order of operations is taught, and they have been taught everything they need to know about it by then
it’s hard to pump so much knowledge into children and teenagers
…and yet have you not noticed that teenagers almost never get this wrong - only adults do
Using “PEMDAS” to argue about the order of operations in mathematics
…is a totally valid thing to do. The problem is people classifying Distribution (Brackets/Parentheses with a coefficient) as “Multiplication”, when there’s literally no multiplication sign
Math notations and conventions evolve exactly like natural languages
No they don’t. Maths is universal
A lot of it is heavily based on historical thanks and work from previous generations
It’s all based on definitions and proofs, which are immutable
There is no definitive norm, standard or convention of notations and order of operations
You can find them in any high school textbook in your country (notation varies by country, but the rules don’t)
some words only appear in half of them (like “implicit multiplication by juxtaposition”)
“implicit multiplication” doesn’t appear in any Maths textbooks
sentences like “I saw the man with the telescope”, because it’s not clear if you saw him through the telescope or saw him holding (or looking through) a telescope
Yes it is clear (as I think I saw someone already point out here)
I saw the man with the telescope - the man has the telescope
I saw the man, with the telescope - I saw the man through a telescope
I saw the man through the telescope - I saw the man through a telescope
it should also be clear why there are no arguments or proofs for any side
But there are proofs! (There you go again with the “there is no…” red flag) Order of operations proof
FACT CHECK 3/5
It’s only a matter of taste and how widespread a convention or notation is
The rules are in every high school Maths textbook. The notation for your country is in your country’s Maths textbooks
There are no arguments or proofs about what definition is correct
1+1=2 by definition (or whatever the notation is in your country). If you write 1+1=3 then that is wrong by definition
I found a lot of explanations online that were either half-assed or just plain wrong
And you seem to have included most of them so far - “implicit multiplication”, “weak juxtaposition”, “conventions”, etc.
You either were taught something wrong or you misremember it.
Spoiler alert: It’s always the latter
IMHO the mnemonics would be better without “division” and “subtraction”, because it would force people to think about it before blindly applying something the wrong way – “PEMA” for example. Parentheses, exponentiation, multiplication, addition
In fact what would happen is now people wouldn’t know in what order to do division and subtraction, having removed them from the mnemonic (and there’s absolutely no reason at all to remove them - you can do everything in the mnemonic order and it works, provided you also obey the left-to-right rule, which is there to make sure you obey left associativity)
parenthesis and exponents students typically don’t learn the order of operations through some mnemonics they remember them through exercise
That’s not true at all. Have you not read through some of these arguments? They’re all full of “Use BEDMAS!”, “Use PEMDAS!”, “It’s PEMDAS not BEDMAS!” - quite clearly these people DID learn order of operations through the mnemonics
trying to remember some random acronyms
There’s no requirement to memorise any acronym - you can always just make up your own if you find that easier! I did that a lot in university to remember things during the exam
they also state to “not use × to express a simple product”
…because a product is a Term, and to insert a x would break it into 2 Terms
A product is the result of a multiplication
The center dot also should not be used to mean a simple product
Exact same reason. They are saying “don’t turn 1 term into 2 terms”. To put that into the words that you keep using, “don’t use weak juxtaposition”
Nobody at the American Physical Society (at least I hope) would say that 6/2×3 equals one, because that’s just bonkers
Because it would break the rule of left associativity (i.e. left to right). No-one is advocating “multiplication before division” where it would violate left to right (usually by “multiplication” they’re actually referring to Terms, and yes, you literally always have to do Terms before Division)
÷ (obelus), : (colon) or / (solidus), but that is not the case and they can be used interchangeably without any difference in meaning. There are no widespread conventions, that would attribute different meanings
Yes there is. Some countries use : for divide, whereas other countries use it for ratio
most standards forbid multiple divisions with inline notation, for example expressions like this 12/6/2
Name one! Give me a reference! There’s nothing forbidding that in Maths (though we would more usually write it as 12/(6x2)). Again, all you have to do is obey left to right
Funny enough all the examples that N.J. Lennes list in his letter use
…Terms. Same as all textbooks do now
and thus his rule could be replaced by
…Terms, the already-existing rule that he apparently didn’t know about (he mentions them, and products, but manages to completely miss what that actually means)
“Something, something, distributive property, something ….”
Something, something, Distributive Law (yes, some people use the wrong name, but in talking about the property, not the law, you’re knocking down a strawman)
The distributive property is just a property that applies to some operations
…and The Distributive Law applies to every bracketed term that has a coefficient, in this case it’s 2(1+2)
It has nothing to do with the order of operations
And The Distributive Law has everything to do with order of operations, since solving Brackets is literally the first step!
I’ve no idea where this idea comes from
Maybe you should’ve asked someone. Hint: textbooks/teachers
because there aren’t any primary sources (at least I wasn’t able to find any)
Here it is again, textbook references, proofs, memes, the works
should be calculated (distributed) first
Bingo! Distribution isn’t Multiplication
6÷2(3). If we follow the strong juxtaposition convention, we must
…distribute the 2, always
It has nothing to do with the 3 being inside parentheses
It has everything to do with there being a coefficient to the brackets, the 2
Those parentheses are only there, because
…it’s a factorised term, and the opposite of factorising is The Distributive Law
the parentheses do not force the multiplication
No, it forces distribution of the coefficient. a(b+c)=(ab+ac)
The parentheses are only there to make it clear that
…it is a factorised term subject to The Distributive Law
we are implicitly multiplying two separate numbers.
They’re NOT 2 separate numbers. It’s a single, factorised term, in the same way that 2a is a single term, and in this case a is equal to (1+2)!
With the context that the engineer is trying to calculate the radius of a circle it’s clear that they meant r=C/(2π)
Because 2π is a single term, by definition (it’s the product of a multiplication), as is r itself, so that should actually be written r=(C/2π)
When symbols for quantities are combined in a product of two or more quantities, this combination is indicated in one of the following ways: ab,a b,a⋅b,a×b
Incorrect. Only the first one is a term/product (not separated by any operators) - the last 2 are multiplications, and the 2nd one is literally meaningless. Space isn’t defined as meaning anything in Maths
Division of one quantity by another is indicated in one of the following ways:
The first is a fraction
The second is a division
The third is also a fraction
The last is a multiplication by a fraction
Creates ambiguity since space isn’t defined to mean anything in Maths. Looks like a typo - was there meant to be a multiply where the space is? Or was there not meant to be a space??
By definition ab-1=a1b-1=(a/b)
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Interesting that Excel sees =6/2(1+2) as an invalid formula and will not calculate it (at least on mobile). =6/2*(1+2) returns 9 because it’s executing the division and multiplication left to right (6/2=3*3=9).
Google Sheets (mobile) does’t like it either and returns an error. =6/2*(1+2) also returns “9”.
Excel and Google are both wrong. In fact, Microsoft excels 😂in this area, with Excel, the Windows calculator, and MathSolver all getting it wrong in different ways! dotnet.social/@SmartmanApps/111164851485070719
It’s not ambiguous, it’s just that correctly parsing the expression requires more precise application of the order of operations than is typical. It’s unclear, sure. Implicit multiplication having higher precedence is intuitive, sure, but not part of the standard as-written order of operations.
I’d really like to know if and how your view on that matter would change once you read the full post. I know it’s very long and a lot of people won’t read it because they “already know” the answer but I’m pretty sure it would shift your perception at least a bit if you find the time to read it.
My opinion hasn’t changed. The standard order of operations is as well defined as a notational convention can be. It’s not necessarily followed strictly in practice, but it’s easier to view such examples as normal deviation from the rules instead of an implicit disagreement about the rules themselves. For example, I know how to “properly” capitalize my sentences too, and I intentionally do it “wrong” all the time. To an outsider claiming my capitalization is incorrect, I don’t say “I am using a different standard,” I just say “Yes, I know, I don’t care.” This is simpler because it accepts the common knowledge of the “normal” rules and communicates a specific intent to deviate. The alternative is to try to invent a new set of ad hoc rules that justify my side, and explain why these rules are equally valid to the ones we both know and understand.
What is the correct answer according to the convention you follow?
I have a masters in math, please do not condescend. I’m fully aware of both interpretations and your overall point and I’ve explained my response.
I still don’t see a number ;-) but you can take a look at the meme to see other people with math degrees shouting at each other.
Sorry your article wasn’t as interesting as you hoped.
