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 :-)
I did (skimmed it, at least) and I liked it. 🙃
What the heck are you all fighting about? It’s BODMAS.
deleted by creator
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 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
why that’s actually ambiguous.
It isn’t actually ambiguous. You have remembered what you were taught in school, unlike the author of the blog post, who manages to write the whole thing without ever once checking a Maths textbook (which would reveal the only correct answer to be 1).
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.
Yes, unfortunately there are some bad teachers around. I vividly remember the one I had in Year 10, who literally didn’t care if we did well or not. I got sick for an extended period that year, and got a tutor - my Maths improved when I had the tutor (someone who actually helped me to learn the material)!
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.
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.
Seems this whole thing is the pedestrian-math-nerd’s equivalent to the pedestrian-grammar-nerd’s arguments on the Oxford comma. At the end of the day it seems mathematical notation is just as flexible as any other facet of written human communication and the real answer is “make things as clear as possible and if there is ambiguity, further clarify what you are trying to communicate.”
Seems this whole thing is the pedestrian-math-nerd’s equivalent to the pedestrian-grammar-nerd’s arguments on the Oxford comma.
Not even remotely similar. Maths rules are fixed. The order of operations rules are at least 400 years old.
mathematical notation is just as flexible as any other facet of written human communication
No, it isn’t. The book “A history of mathematical notation” is in itself more than 100 years old.
Wow neat, and yet the thread was full of people going back and forth about how the equation can be misinterpreted based on how the order of operations can be interpreted. Thanks for your months later input though.
I only just found the thread yesterday. There’s only 1 “interpretation”, and the only back and forth I’ve seen about interpretations is about implicit multiplication, which isn’t a thing, at all - it’s people conflating The Distributive Law and Terms dotnet.social/@SmartmanApps/110925761375035558
So you are saying exactly what I said; people can misinterpret things that other people have written. Good job. Thanks again for stopping by a 3 month old thread about a dumb meme.
So you are saying exactly what I said; people can misinterpret things that other people have written
No, I’m not. They’re “misinterpreting” something that isn’t even a rule of Maths. There’s no way to misinterpret the actual rules, there’s no way to misinterpret the equation. There’s no alternative interpretations of the notation. Someone who didn’t remember the rules literally made up “implicit multiplication”, and then other people argued with them about what that meant. 😂
You look like a real idiot here. I really suggest you actually read the article instead of “scanning” it. You clearly don’t even understand the term “implicit multiplication” if you’re claiming it’s made up. Implicit multiplication is not the controversial part of this equation, which you would know if you read the article and understood what people in this thread are even talking about. Stop spamming your shitty blog and just. Read. The. Article.
read the article instead of “scanning” it.
I stopped reading as soon as I saw the claim that the right answer was wrong. I then scanned it for any textbook references, and there were none (as expected).
You clearly don’t even understand the term “implicit multiplication” if you’re claiming it’s made up
Funny that you use the word “term”, since Terms are ONE of the things that people are referring to when they say “implicit multiplication” - the other being The Distributive Law. i.e. Two DIFFERENT actual rules of Maths have been lumped in together in a made-up rule (by people who don’t remember the actual rules).
BTW if you think it’s not made-up then provide me with a Maths textbook reference that uses it. Spoiler alert: you won’t find any.
Implicit multiplication is not the controversial part of this equation
It’s not the ONLY controversial part of the equation - people make other mistakes with it too - but it’s the biggest part. It’s the mistake that most people have made.
shitty blog
So that’s what you think of people who educate with actual Maths textbook references?
Read. The. Article.
Read Maths textbooks.
I read the whole article. I don’t agree with the notation of the American Physical Society, but who am I to argue that? 😄
I started out thinking I knew how the order of operations worked and ended up with a broader view of the subject. Thank you for opening my mind a bit today. I will be more explicit in my notations from now on.
I don’t agree with the notation of the American Physical Society
I clicked on the link to see what you were talking about, and the quotes which are used in the blog aren’t in there at all. i.e. I searched the whole document, not just the referenced page, and, for example, the expression “multiplication before division” isn’t in there at all. On the other hand the stuff about not inserting multiplication signs into terms is 100% correct, because you are breaking up one term into two, and dropping the precedence from Terms to Multiplication, which changes the answer.
Direct quote from the article:
the American Physical Society state in their Style and Notation Guide on page 21 that they do “multiplication before division”, but you must be careful to not take that out of context
Yep, that’s the “quote” in the blog, but if you click on the link not only is it not on page 21, it’s not in there at all. i.e. the quote - if it even is a quote - is out of context.
Ackshually, the answer is 4
6÷2*(1+2)
6÷(1+2)*2
6÷(3)*2
2*2
4
You’re welcome
psychopath
I recall learning in school that it should be left to right when in doubt. Probably a cop-out from the teacher
Probably a cop-out from the teacher
No, that’s an actual convention of Maths, to make sure people (who don’t know better) obey the actual rule of left associativity.
“when in doubt” is a bit broad but left to right is a great default for operations with the same priority. There is actually a way to calculate in any order if divisions are converted to multiplications (by using the reciprocal value) and subtractions are converted to additions (by negating the value) that requires at least a little bit of math knowledge and experience so it’s typically not taught until later to prevent even more confusion.
For example this: 6 / 2 * 3 can also be rewritten as 6 * 2⁻¹ * 3 and because multiplication is commutative you can now do it in any order for example like 3 * 6 * 2⁻¹
You can also “rearrange” the order without changing the meaning if you move the correct operation (left to the number) with it (should only be done with explicit multiplication)
6 / 2 * 3 into 6 * 3 / 2 (note that I moved the division with the 2)
You can even bring the two to the front. Just remember that left to the six is an “imaginary” (don’t quote me ^^) multiplication. And because we can’t just move “/2” to the beginning we have to insert a one (empty product - check Wikipedia) like so:
1 / 2 * 6 * 3
This also works for addition and subtraction
7 + 8 - 5
You can move them around if you take the operation left to the number with it. With addition the “imaginary” operation at the beginning is a plus sign and the implicit number you use is zero (empty sum - check Wikipedia)
8 - 5 + 7
or like this
0 - 5 + 8 + 7
because with negative numbers you can use the minus sign to indicate negative numbers you can even drop the leading zero like this
-5 + 8 + 7
That’s not really possible with multiplication because “/2” is not a valid notation for “1/2”
6 / 2 * 3
Semi-related: something in me wants to read that as 6 / (2*3), because 6 * 3 / 2 feels like a much more ‘natural’ way to write it
Semi-related: something in me wants to read that as 6 / (2*3)
100% related actually, since that’s the actual next line of working out. i.e. you cannot remove brackets unless there is only 1 term left inside, a mistake which those who have prematurely removed brackets have made and ended up with the wrong answer (because it flips the 3 from being in the denominator to being in the numerator).
6 / 2 * 3 into 6 * 3 / 2 (note that I moved the division with the 2)
And note that it doesn’t work if the multiply was an addition. e.g. 6/2+3=6 but 6+3/2=7.5. Multiplication and division are both binary operators, and you can’t move them around unless you also move the term to the left with it. i.e. 6/2+3=6. 3+6/2=6.
Just remember that left to the six is an “imaginary” (don’t quote me ^^) multiplication
No, to the left of the 6 is an actual plus sign, but we don’t write plus signs if it’s at the start of an expression. +6 and x6 aren’t the same thing at all (and, since x is a binary operator, you couldn’t write just x6 anyway - there would have to be a term to it’s left). No expression ever starts with x6.
That’s not really possible with multiplication because “/2” is not a valid notation for “1/2”
It’s not a valid notation for multiplication either - both multiplication and division are binary operators and must be written with 2 terms.
I really hate the social media discussion about this. And the comments in the past teached me, there are two different ways of learning math in the world.
It’s not taught 2 different ways. It’s taught the same around the world (the mnemonics are different but the rules are the same), there’s just 2 types of people - those who remember the rules and those who don’t. You’ll notice students never get these questions wrong, only adults who’ve forgotten the rules.
Typo in article:
If you are however willing to except the possibility that you are wrong.
Except should be ‘accept’.
Not trying to be annoying, but I know people will often find that as a reason to disregard academic arguments.
A person not knowing the difference in usage between except and accept sounds like a perfectly reasonable reason to disregard their math skills.
Especially when said person keeps making incorrect statements about Maths and ignores completely what is taught in high school.
academic arguments
The “academic arguments” can be ignored since this is actually high school Maths - it’s taught in Year 7-8.
Thank you very much 🫶. No it’s not annoying at all. I’m very grateful not only for the fact that you read the post but also that you took the time to point out issues.
I just fixed it, should be live in a few minutes.
My TI-84 Plus is my holy oracle, I will go with whatever it says.
And then get distracted and play some Doom.
TI calcs give the wrong answer, and it’s in their manual why - they only follow the Primary School rule (“inside the brackets”), not the High School rule which supersedes it (The Distributive Law).
It will give 9, just like my 89 emulator. It treats division like a fraction. For a TI, the entire denominator of a fraction needs to be in parentheses or you get into trouble.
It treats division like a fraction
Which is why it gives the wrong answer.
Also you shouldn’t be adding a dot between the 2 and the brackets - that also changes the answer.
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
