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 :)
You lost me on the section when you started going into different calculators, but I read the rest of the post. Well written even if I ultimately disagree!
The reason imo there is ambiguity with these math problems is bad/outdated teaching. The way I was taught pemdas, you always do the left-most operations first, while otherwise still following the ordering.
Doing this for 6÷2(1+2), there is no ambiguity that the answer is 9. You do your parentheses first as always, 6÷2(3), and then since division and multiplication are equal in ordering weight, you do the division first because it's the left most operation, leaving us 3(3), which is of course 9.
If someone wrote this equation with the intention that the answer is 1, they wrote the equation wrong, simple as that.
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.
A couple of edits (not trying to be rude but people sent to your article are going to be pedantic)
*current beliefs
This sentence needs editing: "They even split the category into two and the make a distinction between implicit multiplication with variables other implicit multiplications."
Thank you for reading the post, and thanks for pointing that out. Should be fixed and live in the next few minutes.
Update: Also fixed that sentence. Thank you so much.
I don't see the problem actually.
==========
1+2= 3
No exponents
Nothing remains
I just finished your article and wow! I'm definitely going to save it and share it the next time I come across another one of those viral problems. It was incredibly thorough and well researched, you clearly put a lot of energy and effort into it and it blew me away. It was really refreshing to see someone articulate themselves so passionately with supporting research. I look forward to reading more of your work!! 👏
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!
Nice write-up.
The order of operations is not part of a holy text that must be blindly followed. If these numbers had units and we knew what quantity we were trying to solve for, there would be no argument whatsoever about what to do. This is a question that never comes up in physics because you can use dimensional analysis to check to see if you did the algebra correctly. Context matters.
I feel like if a blog post presents 2 options and labels one as the "scientific" one... And it is a deserved Label. Then there is probably a easy case to be made that we should teach children how to understand scientific papers and solve the equation in it themselves.
Honestly I feel like it reads better too but that is just me
I recall learning in school that it should be left to right when in doubt. Probably a cop-out from the teacher
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.