6÷2(1+2)
6÷2(1+2)
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 :)
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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.
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's only 1 set of rules, and 2 sets of people - those who follow the rules and those who don't.
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.
"Which ruleset do you consider correct" presupposes, as the comment said, that there are 2 rulesets. There aren't. There's the standard, well known, and simplified model which is taught to kids, and there's the real world, where adults communicate by using context and shared understanding. Picking a side here makes no sense.
When the @onion said there were two different sets of rules, you know as well as I do that they meant strong vs. weak juxtaposition.
You're right that in reality nobody would write an equation like this, and if they did they would usually provide context to help resolve it without resorting to having to guess...
But the point of this post is exactly to point out this hole that exists in the standard order of operations, the drama that has resulted from it, and to shine some light on it.
Picking a side makes no sense only if you have the context to otherwise resolve it... If you were told to solve this equation, and given no other context to do so, you would either have to pick a side or resolve it both ways and give both answers. In that scenario, crossing your arms and refusing to because "it doesn't make sense" would get you nowhere.
In all honesty, I think you're acting like the people who say things like "I've never used algebra, so it was worthless teaching me it as a kid" as though there aren't people who would learn something out of this.
When the @onion said there were two different sets of rules, you know as well as I do that they meant strong vs. weak juxtaposition
Neither of which is a rule in Maths, as had already been pointed out. The 2 relevant rules, both of which never got mentioned in the whole discussion, are The Distributive Law and Terms.
You’re right that in reality nobody would write an equation like this
No, that was wrong - they really would. 2(1+2) is the standard way to write a factorised term. i.e. a(b+c).
hole that exists in the standard order of operations
There's not a hole in the order of operations - there's a hole in people's memories where The Distributive Law and Terms used to be. You'll notice no students ever get this wrong, because they remember all the rules.
given no other context to do so, you would either have to pick a side
That "side" being follow the rules of Maths - works every time. :-)
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.
most scientific calculators are designed to use it
They don't. They use The Distributive Law and Terms.