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 :)
If you are so sure that you are right and already “know it all”, why bother and even read this? There is no comment section to argue.
I beg to differ. You utter fool! You created a comment section yourself on lemmy and you are clearly wrong about everything!
You take the mean of 1 and 9 which is 4.5!
/j
Right, because 5 rounds down to 4.5
@Prunebutt meant 4.5! and not 4.5. Because it’s not an integer we have to use the gamma function, the extension of the factorial function to get the actual mean between 1 and 9 => 4.5! = 52.3428 which looks about right 🤣
🤣 I wasn’t even sure if I should post it on lemmy. I mainly wrote it so I can post it under other peoples posts that actually are intended to artificially create drama to hopefully show enough people what the actual problems are with those puzzles.
But I probably am a fool and this is not going anywhere because most people won’t read a 30min article about those math problems :-)
I did (skimmed it, at least) and I liked it. 🙃
Actually the correct answer is clearly 0.2609 if you follow the order of operations correctly:
6/2(1+2)
= 6/23
= 0.26🤣 I’m not sure if you read the post but I also wrote about that (the paragraph right before “What about the real world?”)
I did read the post (well done btw), but I guess I must have missed that. And here I thought I was a comedic genius
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.
Did you read the blog post?
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.
My argument is specifically that using no separation shows intent for which way to interpret and should not default to weak juxtaposition.
Choosing not to use (6/2)(1+2) implies to me to use the only other interpretation.
There’s also the difference between 6/2(1+2) and 6/2*(1+2). I think the post has a point for the latter, but not the former.
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.
6/2=3
3(1+2)=9
You just did division before brackets, which goes against order of operations rules.
For me to read the whole of 2(1+2) as the denominator in a fraction
You just need to know The Distributive Law and Terms.
Read the linked article
The linked article is wrong. Read this - has, you know, actual Maths textbook references in it, unlike the article.
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.
your response is “Following my logic, there is no confusion!”
That’s because the actual rules of Maths have all been followed, including The Distributive Law and Terms.
there clearly is confusion in the wider world here
Amongst people who don’t remember The Distributive Law and Terms.
The blog does a good job of narrowing down why there’s confusion
The blog ignores The Distributive Law and Terms. Notice the complete lack of Maths textbook references in it?
But it isn’t “correct”
It is correct - it’s The Distributive Law.
it’s one of two standard ways of doing it.
There’s only 1 way - the “other way” was made up by people who don’t remember The Distributive Law and/or Terms (more likely both), and very much goes against the standards.
The ambiguity in the question is
…zero.
Hooray! Correct! Anyone who downvoted or disagrees with this needs to read this instead. Includes actual Maths textbooks references.
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.
This has been my devil’s advocate argument.
While I agree the problem as written is ambiguous
It’s not.
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”.
Agree completely! Notice how they ALWAYS leave out high school Maths teachers and textbooks? You know, the ones who actually TEACH this topic. Always people OTHER THAN the people/books who teach this topic (and so always end up with the wrong conclusion).
while basic education and basic calculators use weak juxtaposition
Literally no-one in education uses so-called “weak juxtaposition” - there’s no such thing. There’s The Distributive Law and Terms, both of which use so-called “strong juxtaposition”. Most calculators do too.
Shouldn’t strong juxtaposition be the precedent and the norm
It is. In fact it’s the rules (The Distributive Law and Terms).
We should start saying weak juxtaposition is wrong
Maths teachers already DO say it’s wrong.
This has been my devil’s advocate argument.
No, this is mostly a Maths teacher argument. You started off weak (saying its ambiguous), but then after that almost everything you said is actually correct - maybe you should be a Maths teacher. :-)
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.
I guess if you wrote it out with a different annotation it would be
6
-‐--------‐--------------
2(1+2)
=
6
-‐--------‐--------------
2×3
=
6
–‐--------‐--------------
6
=1
I hate the stupid things though
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Escape symbols?
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6⁄2(1+2) ⇒ 6⁄2*3 ⇒ 6⁄6 ⇒ 1
You’re more patient than me to go to that trouble! 😂 But yeah, looks good. Just one technicality (and relates to how many people arrive at the wrong answer), the 2x3 should be in brackets. Yes if you had a proper fraction bar it wouldn’t matter, but that’s what’s missing with inline writing, and is compensated for with brackets (and brackets can’t be removed unless there’s only 1 term inside). In your original comment, it does indeed look like 6/(2x3), but, to illustrate the issue with what you wrote, as soon as I quoted it, it now looks like (6/2)x3 in my comment.
I’ve seen a calculator interpret 1 ÷ 2π as ½π which was kinda funny
An e-calculator I’m guessing? (either that or Texas Instruments) Desmos USED TO interpret that correctly, but then they made a change with automatically turning division into fractions and broke it (because if you’ve specified division then it’s not a fraction) dotnet.social/@SmartmanApps/111164851485070719
I believe it was a app , yes
All calculators that are listed in the article as following weak juxtaposition would interpreted it that way.
And they’re all wrong dotnet.social/@SmartmanApps/111164851485070719
The ambiguous ones at least have some discussion around it. The ones I’ve seen thenxouple times I had the misfortune of seeing them on Facebook were just straight up basic order of operations questions. They weren’t ambiguous, they were about a 4th grade math level, and all thenpeople from my high-school that complain that school never taught them anything were completely failing to get it.
I’m talking like 4+1x2 and a bunch of people were saying it was 10.
What if the real answer is the friends we made along the way?
That’d be good, but what I’ve found so far here is a whole bunch of people who don’t like being told the actual facts of the matter! 😂
Nope it’s bedmas since everything is brackets
What the heck are you all fighting about? It’s BODMAS.
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So what does BODMAS sound like to the other side?
samdob
They’re arguing about whether Distribution is Multiplication or not. Spoiler alert: it isn’t, it’s Brackets.
I’d would be great if you find the time to read the post and let me know afterwards what you think. It actually looks trivial as a problem but the situation really isn’t, that’s why the article is so long.
It actually looks trivial as a problem
Because it actually is.
that’s why the article is so long
The article was really long because there were so many stawmen in it. Had you checked a Maths textbook or asked a Maths teacher it could’ve been really short, but you never did either.
I was being facetious. I will try to find the time to read the post, but I know already that the problem isn’t trivial. It involves, above all else, human comprehension, which is a very iffy thing, to say the least.
I would do the mighty parentheses first, and then the 2 that dares to touch the mighty parentheses, finally getting to the run-of-the-mill division. Hence the answer is One.
@wischi “Funny enough all the examples that N.J. Lennes list in his letter use implicit multiplications and thus his rule could be replaced by the strong juxtaposition”.
Weird they didn’t need two made-up terms to get it right 100 years ago.
Indeed Duncan. :-)
his rule could be replaced by the strong juxtaposition
“strong juxtaposition” already existed even then in Terms (which Lennes called Terms/Products, but somehow missed the implication of that) and The Distributive Law, so his rule was never adopted because it was never needed - it was just Lennes #LoudlyNotUnderstandingThings (like Terms, which by his own admission was in all the textbooks). 1917 (ii) - Lennes’ letter (Terms and operators)
In other words…
Funny enough all the examples that N.J. Lennes list in his letter use
…Terms/Products., as we do today in modern high school Maths textbooks (but we just use Terms in this context, not Products).
Hi! Nice blog post. Since you asked for feedback I’ll point out the one thing I didn’t really understand. You explain the difference between the calculators by showing excerpts from the manuals and you highlight that in the first manual, implicit multiplication is prioritised. But the text you underlined only refers to implicit multiplication involving special expressions(?) like pi, e, sqrt or log, and nothing about “regular” implicit multiplication like 2(1+3). So while your photos of the calculator results are great proof that the two models use a different order of operations, to me the manuals were a bit confusing since they did not actually seem to prove your point for the example math problems you are discussing. Or maybe I missed something?
only refers to implicit multiplication involving special expressions(?) like pi, e, sqrt or log, and nothing about “regular” implicit multiplication like 2(1+3)
That was a very astute observation you made there! The fact is, for the very reason you stated, there is in fact no such thing as “implicit multiplication” - it is a term which has been made up by people who have forgotten Terms (the first thing you mentioned) and The Distributive Law (the second thing you mentioned). As you’ve noted., these are 2 different rules, and lumping them together as one brings exactly the disastrous results you might expect from lumping different 2 rules together as one…
See here for explanation of all the various rules, including textbook references and proofs.
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.
I am so glad that nothing I do in life will ever cause this problem to matter to me.
The way I was taught in school, the answer is clearly 1, but I did read the blog post and I understand why that’s actually ambiguous.
Fortunately, I don’t have to care, so will sleep well knowing the answer is 1, and that I’m as correct as anyone else. :-p
It’s not ambiguous, it’s just that correctly parsing the expression requires more precise application of the order of operations than is typical. It’s unclear, sure. Implicit multiplication having higher precedence is intuitive, sure, but not part of the standard as-written order of operations.
I’d really like to know if and how your view on that matter would change once you read the full post. I know it’s very long and a lot of people won’t read it because they “already know” the answer but I’m pretty sure it would shift your perception at least a bit if you find the time to read it.
My opinion hasn’t changed. The standard order of operations is as well defined as a notational convention can be. It’s not necessarily followed strictly in practice, but it’s easier to view such examples as normal deviation from the rules instead of an implicit disagreement about the rules themselves. For example, I know how to “properly” capitalize my sentences too, and I intentionally do it “wrong” all the time. To an outsider claiming my capitalization is incorrect, I don’t say “I am using a different standard,” I just say “Yes, I know, I don’t care.” This is simpler because it accepts the common knowledge of the “normal” rules and communicates a specific intent to deviate. The alternative is to try to invent a new set of ad hoc rules that justify my side, and explain why these rules are equally valid to the ones we both know and understand.
The standard order of operations is as well defined as a notational convention can be.
If it was so well defined, then how did two different sets of rules regarding juxtaposition even come to be?
A well-defined order of operations shouldn’t have a hole that big.Also, @wischi asking you to give the answer as defined by your convention isn’t condescending, they’re asking you to put your money where your mouth is…
Your response certainly felt condescending though, especially since your “explanation” was essentially that anyone who disagrees with the convention you follow is wrong and should feel stupid, and that you needn’t even consider it.
If it was so well defined, then how did two different sets of rules regarding juxtaposition even come to be?
They didn’t - neither of them is a rule of Maths.
There aren’t two different sets of rules. There’s the simple model that’s commonly understood and taught to kids, and there’s the real world where you have context and the dynamics of a conversation and years of experience with communication. One is well defined, the other isn’t.
Them asking me to solve the arithmetic problem is condescending, yes.
My response didn’t say “anyone who disagrees with the convention is stupid.” Here’s condescension for you: please don’t make your reading level my problem. What I said was, there’s an unambiguous way to parse the expression according to the commonly understood order of operations, but it is atypical to pay that much attention to the order of operations in practice. If you think that’s a value judgment, that’s on you-- I was very clear in my example about capitalization, “strictly adhering to the conventional order of operations” is something reasonable people often just don’t care about.
There aren’t two different sets of rules. There’s the simple model that’s commonly understood and taught to kids, and there’s the real world where you have context and the dynamics of a conversation and years of experience with communication. One is well defined, the other isn’t.
And that simple model, well-defined model didn’t properly account for juxtaposition, which is how different fields have ended up with two different ways of interpreting it, i.e. strong vs. weak juxtaposition.
In the real world you simply wouldn’t write any equation out in such a way as to allow misinterpretation like this, but that’s ignoring the elephant in the room…
Which is that the reason viral problems like this still come about and why @wischi went through the effort of writing a rather detailed blog on this is because the order of operations most people are taught doesn’t cover juxtaposition.
Them asking me to solve the arithmetic problem is condescending, yes.
Considering your degree specialisation is in solving arithmetic problems, I don’t see the issue with them asking you to put your money where your mouth is and spit out a number if it’s so easy.
My response didn’t say “anyone who disagrees with the convention is stupid.” Here’s condescension for you: please don’t make your reading level my problem.
Ironic that you tell me to check my reading comprehension right after you misquote me, but nonetheless that is the impression your responses have given off - and you haven’t done anything so far to dispel that impression.
What I said was, there’s an unambiguous way to parse the expression according to the commonly understood order of operations, but it is atypical to pay that much attention to the order of operations in practice.
Yes, and the question everyone is asking you is what is that unambiguous way? Which side of weak or strong juxtaposition do you come out on?
If you think that’s a value judgment, that’s on you-- I was very clear in my example about capitalization, “strictly adhering to the conventional order of operations” is something reasonable people often just don’t care about.
The value judgement was actually more to do with your choice of example, and how you applied that example to this debate. It gave me the distinct impression that you view this debate as not worth having, as anybody who does juxtaposition differently from you is wrong out the gate - and again, your further responses only reinforce my impression of you.
why @wischi went through the effort of writing a rather detailed blog on this is because the order of operations most people are taught doesn’t cover juxtaposition.
The order of operations rules do cover it. Did you not notice that the OP never referenced a single Maths textbook? Because, had that been done, the whole house of cards would’ve fallen down. See my Fact Check posts doing exactly that.
And that simple model, well-defined model didn’t properly account for juxtaposition, which is how different fields have ended up with two different ways of interpreting it, i.e. strong vs. weak juxtaposition.
No, that’s just not what happened. “Strong juxtaposition,” while well-defined, is a post-hoc rationalization. Meaning in particular that people who believe that this expression is best interpreted with “strong juxtaposition” don’t really believe in “strong juxtaposition” as a rule. What they really believe is that communication is subtle and context dependent, and the traditional order of operations is not comprehensive enough to describe how people really communicate. And that’s correct.
Considering your degree specialisation is in solving arithmetic problems
My degree specialization is in algebraic topology.
I don’t see the issue with them asking you to put your money where your mouth is and spit out a number if it’s so easy
The issue is that this question disregards and undermines my point and asks me to pick a side, arbitrarily, that (as I’ve already explained) I don’t actually believe in.
Ironic that you tell me to check my reading comprehension right after you misquote me, but nonetheless that is the impression your responses have given off - and you haven’t done anything so far to dispel that impression.
I didn’t misread, you’re in denial.
Yes, and the question everyone is asking you is what is that unambiguous way? Which side of weak or strong juxtaposition do you come out on?
Hopefully by this point in the comment you understand that I don’t believe the question makes sense.
The value judgement was actually more to do with your choice of example, and how you applied that example to this debate. It gave me the distinct impression that you view this debate as not worth having, as anybody who does juxtaposition differently from you is wrong out the gate - and again, your further responses only reinforce my impression of you.
Again, that’s your fault-- you’ve clearly misinterpreted what I said. If I didn’t think this conversation was worth having I wouldn’t be responding to you.
No, that’s just not what happened. “Strong juxtaposition,” while well-defined, is a post-hoc rationalization. Meaning in particular that people who believe that this expression is best interpreted with “strong juxtaposition” don’t really believe in “strong juxtaposition” as a rule. What they really believe is that communication is subtle and context dependent, and the traditional order of operations is not comprehensive enough to describe how people really communicate. And that’s correct.
I think you’re putting the cart before the horse here - there is definitely a communication issue around juxtaposition, but you’re acting as though strong juxtaposition is some kind of social commentary on the standard order of operations rather than the reality that it is an interpretation that arose due to ambiguity, just as weak juxtaposition did.
If it were people just trying to make a point, then why would it be so widely used and most scientific calculators are designed to use it, or allow its use. This debate exists because so many people ascribe to one or the other without thinking.
My degree specialization is in algebraic topology.
One - that does sound kind of cool
Two - You still have a mathematics degree do you not? You said this was an easy “unambiguous” problem to solve, so I don’t see how you’re prohibited from solving it…
The issue is that this question disregards and undermines my point and asks me to pick a side, arbitrarily, that (as I’ve already explained) I don’t actually believe in.
God saying stuff like that, you sound just like an enlightened centrist…
Anyways, even if you don’t want to comment on the strong vs. weak juxtaposition debate, unless you simply intend on never solving any equation with implicit multiplication by juxtaposition ever again, then you must have a way of interpreting it.
That is what you’re being asked to disclose, since you seem to be very certain that there is a correct way of resolving this. You’ve brought the question upon yourself.
If you don’t want to take a side, simply saying the rules are ambiguous and technically both positions are correct depending on what field you’re in is also a valid position…
But denying the problem all together is not productive.
I didn’t misread, you’re in denial.
In the first place I don’t think you’ve proven me wrong. As far as I can tell your comments still boil down to that you think the whole debate is wrong, and that engaging in the debate is dumb.
But I can say for certain that you either misread or deliberately misconstrued at least part of my reply, because when responding to me you removed the “you follow” from it, which changes the interpretation.
If you think that wasn’t what I said, feel free to go back and look.
Hopefully by this point in the comment you understand that I don’t believe the question makes sense.
I understand you don’t believe the question makes sense, you’ve said that enough times…
But I’ll just refer you to my earlier comment - unless you intend on never solving any equation involving implicit multiplication ever again, then you must ascribe to one way or the other of resolving it.
Again, that’s your fault-- you’ve clearly misinterpreted what I said.
Then tell me how I’ve misinterpreted what you said, because I stick by what I said as far as your example goes.
Your choice of example is not only a much more clear cut issue, being that most kids are taught by primary school (or the US equivalent) how and where to capitalise their letters, and to me it also demonstrates that you’ve not understood that the whole reason this debate is a thing is directly because there’s no “wrong way” of doing this.
If I didn’t think this conversation was worth having I wouldn’t be responding to you.
I understand you see this conversation with me as worth having, but I suspect this is more to do with wanting to resolve this conversation in your favour than it is to do with the underlying debate.
What is the correct answer according to the convention you follow?
I have a masters in math, please do not condescend. I’m fully aware of both interpretations and your overall point and I’ve explained my response.
I still don’t see a number ;-) but you can take a look at the meme to see other people with math degrees shouting at each other.
Sorry your article wasn’t as interesting as you hoped.
The difference is that there are two sets of rules already in use by large groups of people, so which do you consider correct?
There aren’t.
They weren’t asking you if there are two sets of rules, we’re in a thread that’s basically all qbout the Weak vs. Strong juxtaposition debate, they asked you which you consider correct.
Giving the answer to a question they didn’t ask to avoid the one they did is immature.
Ah yes, simply “answer the question with an incorrect premise instead of refuting the premise.” When did you stop beating your wife?
That’s not what they asked me. I have no problem answering questions that are asked in good faith.
I can’t have stopped because I never started, because I’m not even married… See, even I can answer your bad faith question better than you answered the one @onion asked you.
But I will give it to you that my comment should’ve stipulated avoiding reasonable questions.
The difference is that there are two sets of rules already in use by large groups of people, so which do you consider correct?
However I still think you need your eyes checked, as the end of this comment by @onion is very clearly a question asking you WHICH ruleset you consider correct.
Unless you’re refusing the notion of multiplication by juxtaposition entirely, then you must be on one side of this or the other.
There’s only 1 set of rules, and 2 sets of people - those who follow the rules and those who don’t.
