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 :)
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”.
The meme refers to the problem of handling implicit multiplication by juxtaposition.
Depending on what field you're in, implicit multiplication takes priority over explicit multiplication/division (known as strong juxtaposition) rather than what you and a lot of people would assume (known as weak juxtaposition).
With weak juxtaposition you end up 9 just as you did, but with strong juxtaposition you end up with 1 instead.
For most people and most scenarios this doesn't matter, as you'd never encounter such ambiguous equations outside of viral puzzles like this, but it is worth knowing that not all fields agree on how implicit multiplication is handled.
Seems this whole thing is the pedestrian-math-nerd’s equivalent to the pedestrian-grammar-nerd’s arguments on the Oxford comma. At the end of the day it seems mathematical notation is just as flexible as any other facet of written human communication and the real answer is “make things as clear as possible and if there is ambiguity, further clarify what you are trying to communicate.”
Pretty much. While it's worth knowing that not everyone agrees on how implicit multiplication is prioritised, anywhere that everyone agreeing on the answer actually mattered, you wouldn't write an equation as ambiguous as this one in the first place
Just saw the image you posted and it’s awesome :-) I’m part of the group that can’t solve it, because I don’t know the 🌭 function from the top of my head. I also found the choice of symbols interesting that 🌭 is analytical continuation of 🍔 and not the other way round 🤣
I agree with your core message, that the issue is caused by bad notation. However I don’t really see why you consider implicit multiplication to be the sole reason. In my mind, a/bc is equally as ambiguous as a/b*c. The symbols are not important.
You don’t even consider this in your article, instead you seem to take the position that the operations are resolved from left to right. This idea probably comes from programming languages, as they commonly use this convention, but I haven’t seen this defined in mathematics anywhere. I’m open to being wrong here, so if you can show me such a definition from an authoritative source (maybe ISO) I’d be thankful.
As it stands, you basically claim “the original notation is ambiguous, but with explicit × the answer is obviously nine, because my two calculators agree”, even though you just discounted calculator proofs. By the way, both calculators explicitly define this left-to-right order in their documentation.
The ISO section 7.1.3 you quoted is very reasonable and succinct, and contradicts your claim that explicit multiplication sign removes ambiguity. There would be no need for this section if a left-to-right rule existed.
Standards are as mentioned in the article often extra careful to prevent confusion and thus often stricter than widespread conventions with things they allow and don’t allow.
a/b*c is not ambiguous because no widespread convention would treat it any other way than (a/b)*c.
But you can certainly try to proof me wrong by showing me a calculator that would evaluate 6/2*3 to anything but 9.
So if there is not a single calculator out there that would come to a different result, how can it be ambiguous?
Update: Standards are rule-books for real projects to make it simpler to work together. It’s a bit like a Scrabble dictionary. If a word is missing in the official Scrabble dictionary, it doesn’t automatically mean that it’s not a real word, it just means that it wouldn’t be allowed to play that word in official Scrabble tournaments.
Same with (ISO) standards. Just because the standard forbids it doesn’t mean it’s not widespread or forbidden generally. It’s only forbidden in a context where all participants agreed to follow the standard.
All of the programming languages I can think of apply operator precedence as noted in the first reply. That’s the only standard I ever learned, and I’ve never seen any ambiguity in that.
isn’t that division sign I only saw Americans use written like this (÷) means it’s a fraction? so it’s 6÷2, since the divisor (or what is it called in english, the bottom half of the fraction) isn’t in parenthesis, so it would be foolish to put the whole 2(1+2) down there, there’s no reason for that.
so it’s (6/2)(1+2) which is 33 = 9.
the other way around would be 6÷(2(1+2)) if the whole expression is in the divisor and than that’s 1.
tho I’m not really proficient in math, I have eventually failed it in university, but if I remember my teachers correctly, this should be the way. but again, where I live, we never use the ÷ sign, only in elementary school where we divide on paper. instead we use the fraction form, and with that, these kind of seemingly ambiguous expressions doesn’t exist.
I Looked up doing factorials and n! = n(n – 1) is used interchangeably with n! = n*(n – 1)
yeah, the way I have been taught is that either you put the multiplication sign there or not, it’s the exact same, there’s absolutely no difference in n(n-1) and n*(n-1). in the end, you treat it like the * sign is there and it’s just matter of convenience you can leave it off.
Americans use PEMDAS, in the UK we use BODMAS, and I assume other English speaking countries use one or the other, but there is no difference between them in terms of order of operations, because it’s:
brackets (parenthesis)
orders (exponentials)
division & multiplication (multiplication & division), performed left to right
addition & subtraction, performed left to right
People who choose to divide or multiply first because of the acronym have just forgotten that they go together left to right.
“Next, we perform operations on multiplication or division from left to right.” Found that in like 20 sources. Funny how I didn’t see any of that yesterday.
This is crazy swear never heard that rule.
100% sure I solved everything in class following the acronym. Glad we sorted this out before I helped any kids do Homework.
The ÷ sign isn’t used by “Americans”, it’s used by small children. As soon as you learn basic mathematical notation in your introductory algebra class, you’ve outgrown the use of that symbol.
It’s also clearly not a bug as some people suggest. Bugs are – by definition – unintended behavior.
There are plenty of bugs that are well documented. I can’t tell you the number of times that I’ve seen someone do something wrong, that they think is 100% right, and “carefully” document it. Then someone finds an edge case and points out the defined behavior has a bug, because the human forgot to account for something.
The other thing I’d point out that I didn’t see in your blog is that I’ve seen many many people say they need to evaluate the 2(3) portion first because “parenthesis”. No matter how many times I explain that this is a notation for multiplication, they try to claim it doesn’t matter because parenthesis. screams into the void
The fact of the matter is that any competent person that has to write out one of these equations will do so in a way that leaves no ambiguity. These viral math posts are just designed to insert ambiguity where it shouldn’t be, and prey on people who can’t remember middle school math.
Regarding your first part in general true, but in this case the sheer amount of calculators for both conventions show that this is indeed intended behavior.
Regarding your second point I tried to address that in the “distributive property” section, maybe I need to rewrite it a bit to be more clear.
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.
It’s like using literally to add emphasis to something that you are saying figuratively. It’s not objectively “wrong” to do it, but the practice is adding uncertainty where there didn’t need to be any, and thus slightly diminishes our ability to communicate clearly.
Ah, but if you use the rules BODMSA (or PEDMSA) then you can follow the letter order strictly, ignoring the equal precedence left-to-right rule, and you still get the correct answer. Therefore clearly we should start teaching BODMSA in primary schools. Or perhaps BFEDMSA. (Brackets, named Functions, Exponentiation, Division, Multiplication, Subtraction, Addition). I’m sure that would remove all confusion and stop all arguments. … Or perhaps we need another letter to clarify whether implicit multiplication with a coefficient and no symbol is different to explicit multiplication… BFEIDMSA or BFEDIMSA. Shall we vote on it?
I think anything after (whichever grade your country introduces fractions in) should exclusively use fractions or multiplication with fractions to express division in order to disambiguate. A division symbol should never be used after fractions are introduced.
This way, it doesn’t really matter which juxtaposition you prefer, because it will never be ambiguous.
Anything before (whichever grade introduces fractions) should simply overuse brackets.
This comment was written in a couple of seconds, so if I missed something obvious, feel free to obliterate me.
Also PIMDAS (we had this conversation in my class this semester as we had a very wide range of ages and regions present in the class) (I is for indices) (I don’t remember what the Colombian students said, for some reason we had a group of 3 Colombians in our class of 12 nowhere near Colombia)
That said, the question is ambiguously written. Maybe the popularity of this will result in calculators being more consistent with how they interpret implicit multiplication signs.
(my preference is to show two lines, one with the numerator and one with the divisor)
I would also add that you shouldn’t be using a basic calculator to solve multi part problems. Second, I haven’t seen a division sign used in a formal math class since elementary and possibly junior high. These things are almost always written as fractions which makes the logic easier to follow. The entire point of working in convention is so that results are reproducible. The real problem though is that these are not written to educate anyone. They are deliberately written to confuse so that some social media personality can make money from clicks. If someone really wants to practice math skip the click and head over to the Kahn Academy or something similar.
I read the article, and it explained the situation and the resultant confusion very well. That said, could we not have some international body just make a decision one way or the other, instead of perpetuating this uncertainty?
It’s practically impossible to do that because (applied) mathematics is such a diverse field and there is no global authority (and really can’t be).
Math notation is very similar to natural languages what you are proposing is a bit like saying we have an ambiguity in english with the word “bat”. It can mean the animal or the sport device. To prevent confusion the oxford dictionary editors just decide that from now on “bat” only refers to the animal and not the club. Problem solved globally? Probably not :-)
What you can do/try is to enforce some rules in smaller groups, like various style guides and standards are trying to do. For example it’s way simpler for a university to enforce certain conventions and styles for the work they and their students produce. But all engineers in Belgium couldn’t care less what a university in India is thinking about math notations.
For real projects that involve many people there are typically industry standards that are followed that work a bit like in the university example and is enforced by the participants of the project.
You probably missed the part where the article talks about university level math, and that strong juxtaposition is common there.
I also think that many conventions are bad, but once they exist, their badness doesn’t make them stop being used and relied on by a lot of people.
I don’t have any skin in the game as I never ran into ambiguity. My university professors simply always used fractions, therefore completely getting rid of any possible ambiguity.
As an engineer with a full PhD. I’d say we engineers aren’t that great with math problems like this. Thus any responsible engineer would write it in a way that cannot be misinterpreted. Because misinterpreted mathematics can kill people…
I don’t have much to say on this, other than that I appreciate how well-written this deep dive is and I appreciate you for writing it. People get so polarized with these viral math problems and it baffles me.
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