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
🤣 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 :-)
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
I did (skimmed it, at least) and I liked it. 🙃
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’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
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 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.
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
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! 😂
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.
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 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.
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.
Hooray! Correct! Anyone who downvoted or disagrees with this needs to read this instead. Includes actual Maths textbooks references.
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.
@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).
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
deleted by creator
Escape symbols?
deleted by creator
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.
FACT CHECK 5/5
most people just dismiss that, because they “already know” the answer
Maths teachers already know how to do Maths. Huh, who would’ve thought? Next thing you’ll be telling me is English teachers know the rules of grammar and how to spell!
and a two-sentence comment can’t convince them how and why it’s ambiguous
Literally NOTHING can convince a Maths teacher it’s ambiguous - Maths teachers already know all the rules of Maths, and which ones you’re breaking
Why read something if you have nothing to learn about the topic that’s so simple that you know for a fact that you are right
To fact check it for the benefit of others
At this point I hope you understand how and why the original problem is ambiguous
At this point I hope you understand why it isn’t ambiguous. Tip: next time check some Maths textbooks or ask a Maths teacher
that one of the two shouldn’t even be a thing
Neither of them is a thing
not everybody shares your opinion and preferences
Facts you mean. The rules of Maths are facts
There is no mathematically true
There absolutely is! You just chose not to ask any experts about it
the most important thing with this “viral math” expressions is to recognize that
…they are all solvable by following the rules of Maths
One could argue that there should also be a strong connection between coefficients and variables (like in r=C/2π)
There is - The Distributive Law and Terms
it’s fine to stick to “BIDMAS” in school but be aware that that’s not the full story
No, BIDMAS and left to right is the full story
If you encounter such discussions in the wild you could just post a link to this page
No, post a link to this order of operations thread index - it has textbook references, proofs, memes, worked examples, the works!
FACT CHECK 3/5
It’s only a matter of taste and how widespread a convention or notation is
The rules are in every high school Maths textbook. The notation for your country is in your country’s Maths textbooks
There are no arguments or proofs about what definition is correct
1+1=2 by definition (or whatever the notation is in your country). If you write 1+1=3 then that is wrong by definition
I found a lot of explanations online that were either half-assed or just plain wrong
And you seem to have included most of them so far - “implicit multiplication”, “weak juxtaposition”, “conventions”, etc.
You either were taught something wrong or you misremember it.
Spoiler alert: It’s always the latter
IMHO the mnemonics would be better without “division” and “subtraction”, because it would force people to think about it before blindly applying something the wrong way – “PEMA” for example. Parentheses, exponentiation, multiplication, addition
In fact what would happen is now people wouldn’t know in what order to do division and subtraction, having removed them from the mnemonic (and there’s absolutely no reason at all to remove them - you can do everything in the mnemonic order and it works, provided you also obey the left-to-right rule, which is there to make sure you obey left associativity)
parenthesis and exponents students typically don’t learn the order of operations through some mnemonics they remember them through exercise
That’s not true at all. Have you not read through some of these arguments? They’re all full of “Use BEDMAS!”, “Use PEMDAS!”, “It’s PEMDAS not BEDMAS!” - quite clearly these people DID learn order of operations through the mnemonics
trying to remember some random acronyms
There’s no requirement to memorise any acronym - you can always just make up your own if you find that easier! I did that a lot in university to remember things during the exam
they also state to “not use × to express a simple product”
…because a product is a Term, and to insert a x would break it into 2 Terms
A product is the result of a multiplication
The center dot also should not be used to mean a simple product
Exact same reason. They are saying “don’t turn 1 term into 2 terms”. To put that into the words that you keep using, “don’t use weak juxtaposition”
Nobody at the American Physical Society (at least I hope) would say that 6/2×3 equals one, because that’s just bonkers
Because it would break the rule of left associativity (i.e. left to right). No-one is advocating “multiplication before division” where it would violate left to right (usually by “multiplication” they’re actually referring to Terms, and yes, you literally always have to do Terms before Division)
÷ (obelus), : (colon) or / (solidus), but that is not the case and they can be used interchangeably without any difference in meaning. There are no widespread conventions, that would attribute different meanings
Yes there is. Some countries use : for divide, whereas other countries use it for ratio
most standards forbid multiple divisions with inline notation, for example expressions like this 12/6/2
Name one! Give me a reference! There’s nothing forbidding that in Maths (though we would more usually write it as 12/(6x2)). Again, all you have to do is obey left to right
Funny enough all the examples that N.J. Lennes list in his letter use
…Terms. Same as all textbooks do now
and thus his rule could be replaced by
…Terms, the already-existing rule that he apparently didn’t know about (he mentions them, and products, but manages to completely miss what that actually means)
“Something, something, distributive property, something ….”
Something, something, Distributive Law (yes, some people use the wrong name, but in talking about the property, not the law, you’re knocking down a strawman)
The distributive property is just a property that applies to some operations
…and The Distributive Law applies to every bracketed term that has a coefficient, in this case it’s 2(1+2)
It has nothing to do with the order of operations
And The Distributive Law has everything to do with order of operations, since solving Brackets is literally the first step!
I’ve no idea where this idea comes from
Maybe you should’ve asked someone. Hint: textbooks/teachers
because there aren’t any primary sources (at least I wasn’t able to find any)
Here it is again, textbook references, proofs, memes, the works
should be calculated (distributed) first
Bingo! Distribution isn’t Multiplication
6÷2(3). If we follow the strong juxtaposition convention, we must
…distribute the 2, always
It has nothing to do with the 3 being inside parentheses
It has everything to do with there being a coefficient to the brackets, the 2
Those parentheses are only there, because
…it’s a factorised term, and the opposite of factorising is The Distributive Law
the parentheses do not force the multiplication
No, it forces distribution of the coefficient. a(b+c)=(ab+ac)
The parentheses are only there to make it clear that
…it is a factorised term subject to The Distributive Law
we are implicitly multiplying two separate numbers.
They’re NOT 2 separate numbers. It’s a single, factorised term, in the same way that 2a is a single term, and in this case a is equal to (1+2)!
With the context that the engineer is trying to calculate the radius of a circle it’s clear that they meant r=C/(2π)
Because 2π is a single term, by definition (it’s the product of a multiplication), as is r itself, so that should actually be written r=(C/2π)
When symbols for quantities are combined in a product of two or more quantities, this combination is indicated in one of the following ways: ab,a b,a⋅b,a×b
Incorrect. Only the first one is a term/product (not separated by any operators) - the last 2 are multiplications, and the 2nd one is literally meaningless. Space isn’t defined as meaning anything in Maths
Division of one quantity by another is indicated in one of the following ways:
The first is a fraction
The second is a division
The third is also a fraction
The last is a multiplication by a fraction
Creates ambiguity since space isn’t defined to mean anything in Maths. Looks like a typo - was there meant to be a multiply where the space is? Or was there not meant to be a space??
By definition ab-1=a1b-1=(a/b)
FACT CHECK 4/5
a solidus (/) shall not be followed by a multiplication sign or a division sign on the same line
There’s absolutely nothing wrong with doing that. The order of operations rules have everything covered. Anything which follows an operator is a separate term. Anything which has a fraction bar or brackets is a single term
most typical programming languages don’t allow omitting the multiplication operator
Because they don’t come with order of operations built-in - the programmer has to implement it (which is why so many e-calculators are wrong)
“.NET IDE0048 – Add parentheses for clarity”
Microsoft has 3 different software packages which get order of operations wrong in 3 different ways, so I wouldn’t be using them as an example! There are multiple rules of Maths they don’t obey (like always rounding up 0.5)
Let’s say we want to clean up and simplify the following statement … o×s×c×(α+β) … by removing the explicit multiplication sign and order the factors alphabetically: cos(α+β) Nobody in their right mind would remove the explicit multiplication sign in this case
This is wrong in so many ways!
- you did multiplication before brackets, which violates order of operations rules! You didn’t give enough information to solve the brackets - i.e. you left it ambiguous - you can’t just go “oh well, I’ll just do multiplication then”. No, if you can’t solve Brackets then you can’t solve ANYTHING - that is the whole point of the order of oeprations rules. You MUST do brackets FIRST.
- the term (α+β) doesn’t have a coefficient, so you can’t just randomly decide to give it one. It is a separate term from the rest Is there supposed to be more to this question? Have you made this deliberately ambiguous for example?
- if the question is just to simplify, then no simplification is possible. You’ve not given any values to substitute for the pronumerals
- (α+β) is presumably (you’ve left this ambiguous by not defining them) a couple of angles, and if so, why isn’t the brackets preceded by a trig function?
- As it’s written, it just looks like a straight-forward multiplying and adding pronumerals except you didn’t give us any values for the pronumerals meaning no simplfication is possible
- if this was meant to be a trig question (again, you’ve left out any information that would indicate this, making it ambiguous) then you wouldn’t use c, o, or s for your pronumerals - you’ve got a whole alphabet left you can use. Appropriate choice of pronumerals is something we teach in Maths. e.g. C for cats, D for dogs. You haven’t defined what ANY of these pronumerals are, leaving it ambiguous
Nobody will interpret cos(α+β) as a multiplication of four factors
- as originally written it’s 4 terms, not 1 term. i.e. it’s not cos(α+β), it’s actually oxsxxx(α+β), since that can’t be simplified. And yes, that’s 4 terms multiplied!
From those 7 points, we can see this is not a real Maths problem. You deliberately made it ambiguous (didn’t say what any of the pronumerals are) so you could say “Look! Maths is ambiguous!”. In other words, this is a strawman. If you really think Maths is ambiguous, then why didn’t you use a real Maths example to show that? Spoiler alert: #MathsIsNeverAmbiguous hence why you don’t have a real example to illustrate ambiguity
Implicit multiplications of variables with expressions in parentheses can easily be misinterpreted as functions
No they can’t. See previous points. If there is a function, then you have to define what it is. e.g. f(x)=x². If no function has been defined, then f is the pronumeral f of the factorised term f(x), not a function. And also, if there was a function defined, you wouldn’t use f as a pronumeral as well! You have the whole rest of the alphabet left to use. See my point about we teach appropriate choice of pronumerals
So, ambiguity really hides everywhere
No, it really doesn’t. You just literally made up some examples which go against the rules of Maths then claimed “Look! Maths is ambiguous!”. No, it isn’t - the rules of Maths make sure it’s never ambiguous
IMHO it would be smarter to only allow the calculation if the input is unambiguous.
Which is exactly what calculators do! If you type in something invalid (say you were missing a bracket), it would say “syntax error” or something similar
force the user to write explicit multiplications
Are you saying they shouldn’t be allowed to enter factorised terms? If so, why?
force notation that is never ambiguous
We already do
but that would lead to a very convoluted mess that’s hard to read and write
In what way is 6/2(1+2) either convoluted or hard to read? It’s a term divided by a factorised term - simple
providing context that makes it unambiguous
In other words, follow the rules of Maths.
Links about various potentially ambiguous math notations
Spoiler alert: they’re not
“Most ambiguous phrases and notations in maths”
e.g. fx=f(x), which I already addressed. It’s either been defined as a function or as pronumerals, therefore nothing ambiguous
“Absolute value notation is ambiguous”
No, it’s not. |a|b|c| is the absolute value of a, times b, times the absolute value of c… which you would just write as b|ac|. Unlike brackets you can’t have nested absolute values, so the absolute value of (a times the absolute value of b times c) would make no sense, especially since it’s the EXACT same answer as |abc| anyway!
In-line power towers like
Left associativity. i.e. an exponent is associated with the term to its left - solve exponents right to left
People saying “I don’t know how to interpret this” doesn’t mean it’s ambiguous, nor that it isn’t defined. It just means, you know, they need to look it up (or ask a Maths teacher)! If someone says “I don’t know what the word ‘cat’ means”, you don’t suddenly start running around saying “The word ‘cat’ is ambiguous! The word ‘cat’ is ambiguous!” - you just tell them to look it up in a dictionary. In the case of Maths, you look it up in a Maths textbook
Because the actual math is easy almost everybody has an opinion on it
…and any of them which contradict any of the rules of Maths are demonstrably wrong
Most people also don’t know that with weak and strong juxtaposition there are two conflicting conventions available
…and Maths teachers know that both of them are made-up and not real things in Maths
But those mnemonics cover just the basics. The actual real world is way more complicated and messier than “BODMAS”
Nope. The mnemonics plus left to right covers everything you need to know about it
Even people who know about implicit multiplication by juxtaposition dismiss a lot of details
…because it’s not a real thing
Probably because of confirmation bias and/or because they don’t want to invest so much time into thinking about stupid social media posts
…or because they’re a high school Maths teacher and know all the rules of Maths
the actual problem with the ambiguity can’t be explained in a quick comment
Yes it can…
Forgotten rules of Maths - The Distributive Law (e.g. a(b+c)=(ab+ac)) applies to all bracketed Terms, and Terms are separated by operators and joined by grouping symbols
Bam! Done! Explained in a quick comment
