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 :)
Hey this is interesting, thanks for sharing!
Great read! Easy for everyone to understand, but also thorough. I loved the breakdown into the calculators functionality
isn't that division sign I only saw Americans use written like this (÷) means it's a fraction? so it's 6÷2, since the divisor (or what is it called in english, the bottom half of the fraction) isn't in parenthesis, so it would be foolish to put the whole 2(1+2) down there, there's no reason for that.
so it's (6/2)*(1+2) which is 3*3 = 9.
the other way around would be 6÷(2(1+2)) if the whole expression is in the divisor and than that's 1.
tho I'm not really proficient in math, I have eventually failed it in university, but if I remember my teachers correctly, this should be the way. but again, where I live, we never use the ÷ sign, only in elementary school where we divide on paper. instead we use the fraction form, and with that, these kind of seemingly ambiguous expressions doesn't exist.
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!
Nobody will interpret cos(α+β) as a multiplication of four factors
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
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?)
#DontForgetDistribution #MathsIsNeverAmbiguous