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@mc yup, well aware of the wikipedia article. Been over it countless times with people on this debate.
@mc note that the ambiguity only arrises with "division with a slash" which is the wording used even in the exceptions to the general rule such as the journal mentioned.
Splitting hairs (figuratively)
@mc
Interesting. The Wikipedia article's current wording seems very slightly ambiguous here:
"...in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2n equals 1 ÷ (2n), not (1 ÷ 2)n. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[22] and this is also the convention observed in prominent physics textbooks..."
It begins by talking about implied multiplication specifically, but then the example doesn't include the "implied" qualification. Perhaps those publications exclusively use implied multiplication anyway, making the implicit/explicit distinction irrelevant.
@freemo
Splitting hairs (figuratively)
Yea I picked up on that too, it is a bit ambiguous.
I can say from personal expiernce that 1) no one ive ever met has done math that uses in-line division. They always make it stacked (like you do when writing) so any change of ambiguity is removed 2) if it is written on a single line they would never (outside of maybe physics those guys are wacky) consider multiplication and division to have different precedence.
@mc
Being charitable to the Casio, I suppose it's possible that the multiplication is only getting higher precedence due to being implicit, i.e. such that "a / bc" is "a / (b * c)" and not "(a / b) * c". I could maybe accept that, as long as "a / b * c" is correctly interpreted as "(a / b) * c". However, I wouldn't ignore the possibility that the Casio is actually giving higher precedence to multiplication in general, which is what I'd be more worried about.
@mc
Thanks for that link. The real answer is that, although PE(MD)(AS) does require explicit multiplication and division to have the same precedence, it does not define the semantics of *implicit* multiplication. As your linked page mentions about halfway down, there are two interpretations in common use. Any ambiguity from this should be avoided by using explicit multiplication. Most programming languages enforce this by disallowing implicit multiplication at all times.
My guess is that Casio is using PE(MD)(AS) correctly, and is using the "algebra class" interpretation (i.e. implicit multiplication has higher precedence) to make that model of calculator more useful to algebra students.
Meanwhile, the Android calculator would naturally follow the "elementary school" interpretation (i.e. implicit multiplication is the same as explicit multiplication) to appeal to a more general audience.
So, both calculators are correct for their intended purposes.
@mc
Clarification: for some (most?) programming languages, the main motivation for not allowing implicit multiplication might actually be to simplify parsing, with preventing ambiguity being more of an added bonus.
@mc
Julia is an interesting case. Implicit multiplication is only allowed where it doesn't clash with the function syntax, it has higher precedence, and it forbids whitespace. https://docs.julialang.org/en/v1/manual/integers-and-floating-point-numbers/#man-numeric-literal-coefficients
julia> x=9; y=2;
julia> 2x/3y-1
2.0
Meanwhile, the Wolfram Language allows implicit multiplication in general, but apparently with the same precedence as explicit multiplication, and whitespace is required. https://reference.wolfram.com/language/ref/Times.html
I appreciate that both languages have rules about whitespace that make the semantics more obvious. The approach used in Julia seems more useful, though.
@mc the casio on the left is wrong, and most calculators, the vast majority, would disagree with it.