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 :)
You state that the ambiguity comes from the implicit multiplication and not the use of the obelus.
I.e. That 6 ÷ 2 x 3 is not ambiguous
What is your source for your statement that there is an accepted convention for the priority of the iinline obelus or solidus symbol?
As far as I’m aware, every style guide states that a fraction bar (preferably) or parentheses should be used to resolve the ambiguity when there are additional operators to the right of a solidus, and that an obelus should never be used.
Which therefore would make it the division expressed with an obelus that creates the ambiguity, and not the implicit multiplication.
(Rest of the post is great)
In this case it’s actually the absence of sources. I couldn’t find a single credible source that states that ÷ has somehow a different operator priority than / or that :
The only things there are a lot of are social media comments claiming that without any source.
My guess is that this comes from a misunderstanding that the obelus sign is forbidden in a lot of standards. But that’s because it can be confused with other symbols and operations and not because the order of operations is somehow unclear.
What is your source for the priority of the / operator?
i.e. why do you say 6 / 2 * 3 is unambiguous?
Every source I’ve seen states that multiplication and division are equal priority operations. And one should clarify, either with a fraction bar (preferably) or parentheses if the order would make a difference.
Same priority operations are solved from left to right. There is not a single credible calculator that would evaluate “6 / 2 * 3” to anything else but 9.
But I challenge you to show me a calculator that says otherwise. In the blog are about 2 or 3 dozend calculators referenced by name all of them say the same thing. Instead of a calculator you can also name a single expert in the field who would say that 6 / 2 * 3 is anything but 9.
Will you accept wolfram alpha as credible source?
https://mathworld.wolfram.com/Solidus.html
Did you read the blog post? I also quoted the exact same thing.
My apologies, I wasn’t trying to spar with you friend, just trying to understand why a/b*c wouldn’t also be considered ambiguous, particularly since an author could have written a*c/b and removed any doubt.
In your blog post you also quoted ISO
You seemed to speak rather definitively that it’s only ambiguous when combined with implicit multiplication.
I agree that almost all calculators and programming languages will interpret consecutive explicit multiplications and divisions with left-to-right precedence.
But as far as I’m aware no such LTR rule has global agreement in mathematics, I was curious if you found something in your research that says otherwise.
It depends on what you mean by global agreement as there is no single source of truth but the left-to-right rule is pretty much default for multiplications/divisions and additions/subtractions. If you however have inline power notations with “^” symbol they are evaluated right-to-left. There are exceptions but those are typically well known in the industry. For example MathCad also evaluates powers from left to right, which is fairly untypical.
It’s not wrong if you make clear what you are doing. You can for example in a diagram call the axis a and b, not really wrong but pretty untypical if everybody else uses x and y, so you should have at least good reasons when doing it differently.
I would be particularly interested if you found something in a mathematical style guide that recommended an expression like
( a / b ) * c
Should be re-written as
a / b * c
Generally speaking, style guides advise rewriting equations for maximum clarity. Which usually includes a guideline of removing parentheses when their existence isn’t needed to clarify intent.
I believe, and I’m particularly interested to see if you found evidence that my understanding is incorrect, that the LTR convention used by calculators and computer programming languages today exists because a deterministic interpretation is a requirement or the hardware, not because any such convention existed prior to that or has been officially codified one way or the other by any mathematics bodies.
So like, forget division for a sec…
In a mathematics paper, you usually wouldn’t write:
(a + (b + c)) + d
You’d write:
a + b + c + d
(Except perhaps if in your paper the parentheses made it easier to follow how you got to that equation.)
Because in mathematics, it will never matter which order you do additions in, so you should drop the parentheses to improve clarity.
On a computer or a calculator though you might get a different result for those two equations like if a+b overflows your accumulator and c is a negative number, or when these are floating points values with significantly different magnitudes.
I believe english speaking engineers just adopted LTR as the convention for how to interpret it since they had to do something, and the english language is a LTR language. I don’t believe that convention exists outside of the context of computing.
The Wolfram quote and ISO quote in particular that you have in your post imply that an inline division followed by an explicit multiplication is ambiguous as to if it should be interpreted as a compound fraction.
If that’s correct, then it would be the inline division that makes it ambiguous, not the implicit multiplication that makes it ambiguous.
If there’s some source from before computers, or outside of the context of computers forcing a decision. Then your assertion that it is the implicit multiplication causes the ambiguity is correct.
I’m not trying to prove you’re wrong, I’m just genuinely curious which it is. And if you found evidence one way or the other.
The link references “a/bc” not “a/b*c”. The first is ambiguous, the second is not.
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.