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 :)
Build two cases, calculate for both, drag both case through the entirety of both problems, get two answers, make a case for both answers, end up with two hypothesis. Easy!
I’m not sure if you read the post yet but I also have a short section about alternative notations which are less ambiguous or never ambiguous. RPN has the same issue as most notations that are never ambiguous namely that it’s hard to read - especially for big expressions.
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 think this speaks to why I have a total of 5 years of college and no degree.
Starting at about 7th grade, math class is taught to every single American school child as if they’re going to grow up to become mathematicians. Formal definitions, proofs, long sets of rules for how you manipulate squiggles to become other squiggles that you’re supposed to obey because that’s what the book says.
Early my 7th grade year, my teacher wrote a long string of numbers and operators on the board, something like 6 + 4 - 7 * 8 + 3 / 9. Then told us to work this problem and then say what we came up with. This divided us into two groups: Those who hadn’t learned Order of Operations on our own time who did (six plus four is ten, minus seven is three, times eight is 24, plus three is 27, divided by nine is three) Three, and who were then told we were wrong and stupid, and those who somehow had, who did (seven times eight is 56, three divided by nine is some tiny fraction…) got a very different number, and were told they were right. Terrible method of teaching, because it alienates the students who need to do the learning right off the bat. And this basically set the tone until I dropped out of college for the second time.
Interesting, I didn’t know about strong implicit multiplication. So I would have said the result is 9. All along my studies in France, up to my physics courses at University, all my teachers used weak implicit multiplication. Could be it’s the norm in France, or they only use it in math studies at University.
I didn’t know until now that I unconsciously use strong implicit multiplication (meaning that I get the answer “1”). I believe it happens more or less as a consequence of starting inside the parentheses and then working my way out.
It is a funny little bit of notational ambiguity, so it is funny that people get riled up about it.
In a scientific context it’s actually very rare to run into that issue because divisions are mostly written as fractions which will completely mitigate the issue.
The strong implicit multiplication will only cause ambiguity after a division with inline notation. Once you use fractions the ambiguity vanishes.
In practice you also rarely see implicit multiplications between numbers but mostly between variables or variables and their coefficients.
Def not a math major (BS/PharmD), but your explanation was like seeing through a visual illusion for the first time! lol
I was always taught PEMDAS growing up, and that the MD and the AS was read left to right in an equation like above. But stating the division as a fraction completely changes my mind now about how this calculation works. I think what would happen in a calculation I use every day if the former was used.
Example: Cockcroft-Gault Equation (estimation of renal function)
(140-age)(kg) / 72(SCr) vs (140-age) X kg ➗72 X SCr
In the first eq (correct one) an 80yo patient who weighs 65kg and has an SCr ~ 1.5 = 36.11
In the latter it = 81.25 (waaay too high for an 80yo lol)
I really hate the social media discussion about this. And the comments in the past teached me, there are two different ways of learning math in the world.
True, and it’s not only about learning math but that there is actually no consensus even amongst experts, about the priority of implicit multiplications (without explicit multiplication sign). In the blog post there are a lot of things that try to show why and how that’s the case.
I read the whole article. I don’t agree with the notation of the American Physical Society, but who am I to argue that? 😄
I started out thinking I knew how the order of operations worked and ended up with a broader view of the subject. Thank you for opening my mind a bit today. I will be more explicit in my notations from now on.
Thank you so much for taking the time. I’m also not convinced that APS’s notation is a very good choice but I’m neither american nor a physisist 🤣
I’d love to see how the exceptions work that the APS added, like allowing explicit multiplications on line-breaks, if they still would do the multiplication first, but I couldn’t find a single instance where somebody following the APS notation had line-break inside an expression.
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.
The expression 6/2(1+2) involves both multiplication and division. According to the order of operations (PEMDAS/BODMAS), you should perform operations inside parentheses first, then any multiplication or division from left to right.
Certainly! The expression 6/2(1+2) is ambiguous due to the implicit multiplication. Let’s solve it in both ways:
Implicit multiplication with higher priority:
[ \frac{6}{2}(1+2) ]
First, solve the parentheses:
[ \frac{6}{2}(3) ]
Now, perform the division:
[ 3 \times 3 = 9 ]
Implicit multiplication with the same priority as division:
[ \frac{6}{2(1+2)} ]
Again, solve the parentheses:
[ \frac{6}{2(3)} ]
Now, perform the multiplication first:
[ \frac{6}{6} = 1 ]
So, depending on the interpretation of implicit multiplication, you can get different results: 9 or 1.
I think it’s funny that ChatGPT figured out 1 and 9 but has the steps completely backwards. First it points out what has high priority and then does the exact opposite, both times 🤣
Forgot the algebra using fruit emoji or whatever the fuck.
Bonus points for the stuff where suddenly one of the symbols has changed and it's "supposedly" 1/2 or 2/3 etc. of a banana now, without that symbol having been defined.
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.
The full story is actually more nuanced than most people think, but the post is actually very long (about 30min) so thank you in advance if you really find the time to read it.
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