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 :)
I found a few typos. In the 2nd paragraph under the section “strong feelings”, you use “than” when it should be “then”. More importantly, when talking about distributive properties, you say x(x+z)=xy+xz. I believe you meant x(y+z)=xy+xz.
Otherwise, I enjoyed that read. I’m embarrassed to say that I did think pemdas meant multiplication came before division, however I’m proud to say that I’ve unconsciously known that it’s important to avoid the ambiguity by putting parentheses everywhere for example when I make formulas in spreadsheets. Which by the way, spreadsheets generally allow multiplication by juxtaposition.
It’s actually fine to do multiplication before division, you just have to be sure about which numbers are intended to be included in the divisor of your fraction!
Thank you so much for taking the time and reading the post. I just fixed the typos, many thanks for pointing them out.
There is nothing really to be embarrassed about and if you look at the comment sections of such viral math posts you can see that you are certainly not the only one. I think that mnemonics that use “MD” and “AS” without grouping like in “PE(MD)(AS)” are really to blame here.
An alternative would be to drop the inverse and only use say multiplication and addition as I suggested with “PEMA” but with “PEMDAS” one basically sets up students for the problem that they think that multiplication comes before division.
Damn ragebait posts, it’s always the same recycled operation. They could at least spice it up, like the discussion about absolute value. What’s |a|b|c|?
What I gather from this, is that Geogebra is superior for not allowing ambiguous notation to be parsed 👌
Your example with the absolute values is actually linked in the “Even more ambiguous math notations” section.
Geogebra has indeed found a good solution but it only works if you input field supports fractions and a lot of calculators (even CAS like WolframAlpha) don’t support that.
Yeah! That’s why I mentioned it, it was a fresh ambiguous notation problem that I’ve never encountered before. Discussions of “is it 1 or 9” get tiring quickly.
At least WA and others tell you how they interpret the input, instead of being a black box (until you get to the manuals). Even though it is obvious in hindsight, I didn’t get why two calculators would yield different results; thanks!
Honestly, I do disagree that the question is ambiguous. The lack of parenthetical separation is itself a choice that informs order of operations. If the answer was meant to be 9, then the 6/2 would be isolated in parenthesis.
It’s covered in the blog, but this is likely due to a bias towards Strong Juxtaposition rules for parentheses rather than Weak. It’s common for those who learned math into advanced algebra/ beginning Calc and beyond, since that’s the usual method for higher math education. But it isn’t “correct”, it’s one of two standard ways of doing it. The ambiguity in the question is intentional and pervasive.
I don’t know what you want, man. The blog’s goal is to describe the problem and why it comes about and your response is “Following my logic, there is no confusion!” when there clearly is confusion in the wider world here. The blog does a good job of narrowing down why there’s confusion, you’re response doesn’t add anything or refute anything. It’s just… you bragging? I’m not certain what your point is.
None of this has a point. We’re talking over a shitpost rant about common use of math symbols. Even the conclusion boils down to it being a context dependent matter of preference. I’m just disagreeing that the original question as posed should be interpreted with weak juxtaposition.
I originally had the same reasoning but came to the opposite conclusion. Multiplication and division have the same precedence, so I read the operations from left to right unless noted otherwise with parentheses. Thus:
6/2=3
3(1+2)=9
For me to read the whole of 2(1+2) as the denominator in a fraction I would expect it to be isolated in parentheses: 6/(2(1+2)).
Reading the blog post, I understand the ambiguity now, but i’m still fascinated that we had the same criticism (no parentheses implies intent) but had opposite conclusions.
It’s hilarious seeing all the genius commenters who didn’t read the linked article and are repeating all the exact answers and arguments that the article rebuts :)
I’m still not used to having combined image and text posts so I usually don’t notice the text portion if it isn’t a big ol’ wall and I hope I’m not the only one.
❤️ True, but I think one of the biggest problems is that it’s pretty long and because you can’t really sense how good/bad/convining the text is it’s always a gamble for everybody if it’s worth reading something for 30min just to find out that the content is garbage.
I hope I did a decent job in explaining the issue(s) but I’m definitely not mad if someone decides that they are not going to read the post and still comment about it.
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)
My years out of school has made me forget about how division notation is actually supposed to work and how genuinely useless the ÷ and / symbols are outside the most basic two-number problems. And it’s entirely me being dumb because I’ve already written problems as 6÷(2(1+2)) to account for it before. Me brain dun work right ;~;
While I agree the problem as written is ambiguous and should be written with explicit operators, I have 1 argument to make. In pretty much every other field if we have a question the answer pretty much always ends up being something along the lines of “well the experts do this” or “this professor at this prestigious university says this”, or “the scientific community says”. The fact that this article even states that academic circles and “scientific” calculators use strong juxtaposition, while basic education and basic calculators use weak juxtaposition is interesting. Why do we treat math differently than pretty much every other field? Shouldn’t strong juxtaposition be the precedent and the norm then just how the scientific community sets precedents for literally every other field? We should start saying weak juxtaposition is wrong and just settle on one.
I tried to be careful to not suggest that scientist only use strong juxtaposition. They use both but are typically very careful to not write ambiguous stuff and practically never write implicit multiplications between numbers because they just simplify it.
At this point it’s probably to late to really fix it and the only viable option is to be aware why and how this ambiguous and not write it that way.
As stated in the “even more ambiguous math notations” it’s far from the only ambiguous situation and it’s practically impossible (and not really necessary) to fix.
Scientist and engineers also know the issue and navigate around it. It’s really a non-issue for experts and the problem is only how and what the general population is taught.
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.
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.
This is not a math problem but a calculator engineering problem. Some solve the sub operations from right to left while other do it from left to right.
It’s not really a calculator engineering problem. If you don’t have time to read the entire blog you should definitely check out the section “But my calculator says…”. It’s actually about order of operations regarding implicit multiplication.
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