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.
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 🤣
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.
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.
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.
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.
Kinda. You can’t define a name, but you can get the compiler to interpret literals as a function. If you have a Num instance for (Integer -> Integer) where,
fromInteger i = x -> x * i
the compiler can interpret integer literals as functions like so
I always hate any viral math post for the simple reason that it gives me PTSD flashbacks to my Real Analysis classes.
The blog post is fine, but could definitely be condensed quite a bit across the board and still effectively make the same points would be my only critique.
At it core Mathematics is the language and practices used in order to communicate numbers to one another and it’s always nice to have someone reasonably argue that any ambiguity of communication means that you’re not communicating effectively.
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