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 disagree. Without explicit direction on OOO we have to follow the operators in order.
The parentheses go first. 1+2=3
Then we have 6 ÷2 ×3
Without parentheses around (2×3) we can't do that first. So OOO would be left to right. 9.
In other words, as an engineer with half a PhD, I don't buy strong juxtaposition. That sounds more like laziness than math.
How are people upvoting you for refusing to read the article?
Because those people also didn't read the article and are reacting from their gut.
As was the person who wrote the article. Did you not notice the complete lack of Maths textbooks in it?
I did read the article. I am commenting that I have never encountered strong juxtaposition and sharing why I think it is a poor choice.
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.
This is high school level Maths. It's not taught at university.
There's "strong juxtaposition" in both Terms and The Distributive Law - you've never encountered either of those?
Because as a high school Maths teacher as soon as I saw the assertion that it was ambiguous I knew the article was wrong. From there I scanned to see if there were any Maths textbooks at any point, and there wasn't. Just another wrong article.
Lol. Read it.
Why would I read something that I know is wrong? #MathsIsNeverAmbiguous
Mathematical notation however can be. Because it's conventions as long as it's not defined on the same page.
Nope. Different regions use different symbols, but within those regions everyone knows what each symbol is, and none of those symbols are in this question anyway.
The rules can be found in any high school Maths textbook.
Let's do a little plausibility analysis, shall we? First, we have humans, you know, famously unable to agree on an universal standard for anything. Then we have me, who has written a PhD thesis for which he has read quite some papers about math and computational biology. Then we have an article that talks about the topic at hand, but that you for some unscientific and completely ridiculous reason refuse to read.
Let me just tell you one last time: you're wrong, you should know that it's possible that you're wrong, and not reading a thing because it could convince you is peak ignorance.
I'm done here, have a good one, and try not to ruin your students too hard.
And yet the order of operations rules have been agreed upon for at least 100 years, possibly at least 400 years.
The fact that I saw it was wrong in the first paragraph is a ridiculous reason to not read the rest?
And let me point out again you have yet to give a single reason for that statement, never mind any actual evidence.
You know proofs, by definition, can't be wrong, right? There are proofs in my thread, unless you have some unscientific and completely ridiculous reason to refuse to read - to quote something I recently heard someone say.
My students? Oh, they're doing good. Thanks for asking! :-) BTW the test included order of operations.
Just read the article. You can't prove something with incomplete evidence. And the article has evidence that both conventions are in use.
If something is disproven, it's disproven - no need for any further evidence.
BTW did you read my thread? If you had you would know what the rules are which are being broken.
I'm fully aware that some people obey the rules of Maths (they're actual documented rules, not "conventions"), and some people don't - I don't need to read the article to find that out.
Notation isn't semantics. Mathematical proofs are working with the semantics. Nobody doubts that those are unambiguous. But notation can be ambiguous. In this case it is: weak juxtaposition vs strong juxtaposition. Read the damn article.
Read it. Was even worse than I was expecting! Did you not notice that a blog about the alleged ambiguity in order of operations actually disobeyed order of operations in a deliberately ambiguous example? I wrote 5 fact check posts about it starting here - you're welcome.
Look, this is not the only case where semantics and syntax don't always map, in the same way e.g.: https://math.stackexchange.com/a/586690
I'm sure it's possible that all your textbooks agree, but if you e.g. read a paper written by someone who isn't from North America (or wherever you're from) it's possible they use different semantics for a notation that for you seems to have clear meaning.
That's not a controversial take. You need to accept that human communication isn't as perfectly unambiguous as mathematics (writing math down using notation is a way of communicating)
Syntax varies, semantics doesn't. e.g. in some places colon is used for division, in others an obelus, but regardless of which notation you use, the interpretation of division is immutable.
They might use different notation, but the semantics is universal.
Writing Maths notation is a way of using Maths, and has to be interpreted according to the rules of Maths - that's what they exist for!
No, you can't prove that some notation is correct and an alternative one isn't. It's all just convention.
Maths is pure logic. Notation is communication, which isn't necessarily super logical. Don't mix the two up.
I never said any of it wasn't correct. It's all correct, just depends on what notation is used in your country as to what's correct in your country.
No, it's all defined. In Australia we use the obelus, which by definition is division. In European countries they use colon, which by definition in those countries means division. 1+1=2 by definition. If you wanna say 1+1=2 is just a convention then you don't understand how Maths works at all.
What you are saying is like saying "there's no such things as dictionaries, there are no definitions, only conventions".
Don't mix up super logical Maths notation with "communication" - it's all defined (just like words which are used to communicate are defined in a dictionary, except Maths definitions don't evolve - we can see the same definitions being used more than 100 years ago. See Lennes' letter).
Yeah, and when you read a paper that contains math, you won't see a declaration about what country’s notation is used for things that aren't defined. So it's entirely possible that you don't know how some piece of notation is supposed to be interpreted immediately.
Of course if there's ambiguity like that, only one interpretation is correct and it should be easy to figure out which one, but that's not guaranteed.
Not hard to work out. It'll be , for decimal point and : for division, or . for decimal point and ÷ or / for division, and those 2 notations never get mixed with each other, so never any ambiguity about which it is. The question here is using ÷ so there's no ambiguity about what that means - it's a division operator (and being an operator, it is separating the terms).
Correct, the definitions and the rules define the semantics.
...the rules of Maths. In fact, when we are first teaching proofs to students we tell them they have to write next to each step which rule of Maths they have used for that step.
Apparently a lot of people do! But yes, unambiguous, and therefore the article is wrong.
Nope. An obelus means divide, and "strong juxtaposition" means it's a Term, and needs The Distributive Law applied if it has brackets.
There is no such thing as weak juxtaposition. That is another reason that the article is wrong. If there is any juxtaposition then it is strong, as per the rules of Maths. You're just giving me even more ammunition at this point.
You just gave me yet another reason it's wrong - it talks about "weak juxtaposition". Even less likely to ever read it now - it's just full of things which are wrong.
How about read my damn thread which contains all the definitions and proofs needed to prove that this article is wrong? You're trying to defend the article... by giving me even more things that are wrong about it. 😂
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...
The oxford comma approach, I agree.
Yay for a voice of reason! I've yet to see anyone who says they have a Ph.D. get this correct (I'm a high school Maths teacher/tutor - I actually teach this topic).
Yeah, but implicit multiplication without a sign is often treated with higher priority.
Sure. That doesn't mean it's right to do.
Please read the article, that's exactly what it's about. There is no right answer.
Let them fight.
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.
There's no decision to be made. The correct rules are already taught in literally every Year 7-8 Maths textbook.
There is a right answer. Read this instead dotnet.social/@SmartmanApps/110897908266416158
Is it though? I've only ever seen it treated as standard multiplication.
Read TFA
Go read the article, it's about you
The article is wrong dotnet.social/@SmartmanApps/110897908266416158
But there is parentheses around (2x3). a(b+c)=(ab+ac) - The Distributive Law. You can't remove them unless there is only 1 term left inside. You removed them when you still had 2 terms inside, 2x3.
6/2(1+2)=6/2(3)=6/(2*3)=6/6=1
OR
6/2(1+2)=6/(2+4)=6/6=1