Starting a new comment thread (I gave up on reading all of them). I'm a high school Maths teacher/tutor. You can read my Mastodon thread about it at Order of operations thread index (I'm giving you the link to the thread index so you can just jump around whichever parts you want to read without having to read the whole thing). Includes Maths textbooks, historical references, proofs, memes, the works.
And for all the people quoting university people, this topic (order of operations) is not taught at university - it is taught in high school. Why would you listen to someone who doesn't teach the topic? (have you not wondered why they never quote Maths textbooks?)
I'm curious if you actually read the whole (admittedly long) page linked in this post, or did you stop after realizing that it was saying something you found disagreeable?
I’m a high school Maths teacher/tutor
What will you tell your students if they show you two different models of calculator, from the same company, where the same sequence of buttons on each produces a different result than on the other, and the user manuals for each explain clearly why they're doing what they are? "One of these calculators is just objectively wrong, trust me on this, #MathsIsNeverAmbiguous" ?
The truth is that there are many different math notations which often do lead to ambiguities.
In the case of the notation you're dismissing in your (hilarious!) meme here, well, outside of anglophone high schools, people don't often encounter the obelus notation for division at all except for as a button on calculators. And there its meaning is ambiguous (as clearly explained in OP's link).
You know EXACTLY where I said those things, and you've been avoiding addressing them ever since because you know they prove the point that #MathsIsNeverAmbiguous See ya.
but the order of operations it’s not well defined with respect to regular explicit multiplication
The only type of multiplication there is is explicit. Neither Terms nor The Distributive Law is classed as "multiplication"
There is no single clear norm or convention
There is a single, standard, order of operations rules
Also, see my thread about people who say there is no evidence/proof/convention - it almost always ends up there actually is, but they didn't look (or didn't want you to look)
The reason why so many people disagree is that
...they have forgotten about Terms and/or The Distributive Law, and are trying to treat a Term as though it's a "multiplication", and it's not. More soon
conflicting conventions about the order of operations for implied multiplication
Let me paraphrase - people disagree about made-up rule
Weak juxtaposition
There's no such thing - there's either juxtaposition or not, and if there is it's either Terms or The Distributive Law
construct “viral math problems” by writing a single-line expression (without a fraction) with a division first and a
...factorised term after that
Note how none of them use a regular multiplication sign, but implicit multiplication to trigger the ambiguity.
There's no ambiguity...
multiplication sign - multiplication
brackets with no multiplication sign (i.e. a coefficient) - The Distributive Law
no multiplication sign and no brackets - Terms (also called products by some. e.g. Lennes)
If it’s a school test, ask you teacher
Why didn't you ask a teacher before writing your blog? Maths tests are only ever ambiguous if there's been a typo. If there's no typo's then there's a right answer and wrong answers. If you think the question is ambiguous then you've not studied enough
maybe they can write it as a fraction to make it clear what they meant
This question already is clear. It's division, NOT a fraction. They are NOT the same thing! Terms are separated by operators and joined by grouping symbols. 1÷2 is 2 terms, ½ is 1 term
you should probably stick to the weak juxtaposition convention
You should literally NEVER use "weak juxtaposition" - it contravenes the rules of Maths (Terms and The Distributive Law)
strong juxtaposition is pretty common in academic circles
...and high school, where it's first taught
(6/2)(1+2)=9
If that was what was meant then that's what would've been written - the 6 and 2 have been joined together to make a single term, and elevated to the precedence of Brackets rather than Division
written in an ambiguous way without telling you what they meant or which convention to follow
You should know, without being told, to follow the rules of Maths when solving it. Voila! No ambiguity
to stir up drama
It stirs up drama because many adults have forgotten the rules of Maths (you'll find students get this right, because they still remember)
Calculators are actually one of the reasons why this problem even exists in the first place
No, you just put the cart before the horse - the problem existing in the first place (programmers not brushing up on their Maths first) is why some calculators do it wrong
“line-based” text, it led to the development of various in-line notations
Yes, we use / to mean divide with computers (since there is no ÷ on the keyboard), which you therefore need to put into brackets if it's a fraction (since there's no fraction bar on the keyboard either)
With most in-line notations there are some situations with conflicting conventions
More often than not even the same manufacturer uses different conventions
According to this video mostly not these days (based on her comments, there's only Texas Instruments which isn't obeying both Terms and The Distributive Law, which she refers to as "PEJMDAS" - she didn't have a manual for the HP calcs). i.e. some manufacturers who were doing it wrong have switched back to doing it correctly
P.S. she makes the same mistake as you, and suggests showing her video to teachers instead of just asking a teacher in the first place herself (she's suggesting to add something to teaching which we already do teach. i.e. ab=(axb)).
none of those two calculators is “wrong”
ANY calculator which doesn't obey all the rules of Maths is wrong!
Bugs are – by definition – unintended behaviour. That is not the case here
So a calculator, which has a specific purpose of solving Maths expressions, giving a wrong answer to a Maths expression isn't "unintended behaviour"? Do go on
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
Neither is ambiguous. #MathsIsNeverAmbiguous ab=(axb) by definition. Here it is referred to in Cajori nearly 100 years ago (1928), and literally every textbook example quoted by Lennes (1917) follows the same definition, as do all modern textbooks. Did you not notice that the blog didn't refer to any Maths textbooks? Nor asked any Maths teachers about it.
[…] the question is ambiguous. There is no right or wrong if there are different conflicting rules. The only ones who claim that there is one rule are the ones which are wrong!
Yeah nah. Actual Maths textbooks and proofs - did you not notice the complete lack of references to textbooks in the blog? It's funny that he mentions Cajori though, given Cajori has a direct reference to Terms #MathsIsNeverAmbiguous
6÷2(1+2) (programming.dev)
https://zeta.one/viral-math/...
Glitch in the matrix (ani.social)