So the answer to 6 ÷ 2(1+2) as presented in the OP is insufficient information provided.
True, although the reference to “author and reader” might imply that we’re talking about published work. Most of the situations involving ambiguity or competing conventions seem to be those where someone has typed an expression on a calculator or computer keyboard, as opposed to those involving either handwritten or typeset mathematical expressions.
As a point of reference, when I type “1/x/y” into MATLAB, I know the program will treat it as “1/(xy)" but for the sake of any non-me readers (or even for the sake of next-month-me), I try to make the effort to write "1/(xy)” in the first place.
Well, yeah, more or less, but don’t expect that to get you anywhere on the facebook thread.
Related, if you type 1/xy into Wolfram Alpha, it writes it as 1 in the numerator, and xy in the denominator, not 1/x times y. So there are multiple rational ways to look at it. If I saw “1/xy” I would think 1 over xy is what is meant and the person wrote it that way because of the limitations of writing everything horizontally on an ASCII text-based platform. The parentheses would disambiguate, but I can see someone thinking it’s obvious what was meant.
Pretty much anyone I know would assume it’s 1/(xy) or be annoyed by the ambiguity and ask for clarification.
The convention that implied multiplication has a different precedence than explicit multiplication is fairly strong.
I’m curious where your strong feelings about PEMDAS are coming from. Are you a mathematician?
To save everybody having to click through that link, Wolfram Alpha parses it as
(6/2) * (1+2) = 9.
In other words it does not give the implied multiplication priority.
Which does not surprise me for typed input.
A thing many of us are sorta missing is that all math notation was originally created for handwriting such that the 2D spatial relationships of the symbols all matter. Relative position (and size) conveys meaning. When we blindly crush that down to a one-dimensional line of uniformly-sized text input, some information is lost and must be supplied by other means, e.g. parentheses, or else ambiguity (or outright change of meaning) is introduced.
Yeah, that’s a more clear way of expressing what I was trying to say in my post with the 1/xy example: I would think 1 over xy is what is meant and the person wrote it that way because of the limitations of writing everything horizontally on an ASCII text-based platform. Now why Wolfram Alpha interprets that as 1/(xy) while the phrase in question is interpreted differently is beyond me.
If I needed to write contained to a linear format in my notes, say, the expression 6÷2(1+2) would mean 6÷(2(1+2)) else I would have written 6÷2 * 2(1+2) or 6÷2 ⋅ 2(1+2) or, even more likely, 6/2 ⋅ 2(1+2). It would be silly, in my opinion, to write it as the OP. When I see something in the format a(b+c), my instinct is to treat that as a single unit, as it looks like it comes out of an equation written as such, not tease out the 2 as part of the preceding division. I do know PEDMAS well, but I would have thought this an error or unintentional ambiguity in transcription.
per wolfram alpha:
1/xy = 1/(x*y)
1/x y = y/x
1 / x * y = y/x
1/xy^2 = (y^2)/x
so it appears when wolfram alpha is dealing with double variables without a space, there’s an implied parenthesis around them that takes precedence prior to division, but not exponents.
That’s using “/” as division. What happenes when it’s “÷”?
I don’t see any changes in the above 4 examples when “÷” is substituted for “/” .
This makes much sense. But replacing the y with (y+1) and expecting the order of evaluation still to be clear is just sadistic.
What’s most desirable is CONSISTENCY, that different implementations use the SAME rules. In most cases all the systems derived from the C language, along with Google and Wolfram, use the same rules. 1/2/3 for example is treated as (1/2)/3
I see that this left-to-right tie-breaking does not apply for exponentiation. Both Google and Wolfram treat 2^3^2 as 2^(3^2). But do NOT blame the C language for this inconsistency: C doesn’t offer an exponentiation primitive.
That’s the standard for anything that deals with exponentiation, because (2^3)^2 is the same thing as the simpler 2^(3*2).
I love a good Douglas Adams reference!
Oh yes it is. In most classrooms.
Really? Even back in 1989, I was taught that multiplication and division was on the same level of order as well as addition and subtraction. I don’t recall being explicitly even taught the PEDMAS acronym, just the order of operations.
Teachers assume students will understand
roots are at the same level as exponents even though it is not PERMDAS
students will know multiplication and division are the same level
students will know addition and subtraction are the same level
But not only do many teachers not teach those caveats, but even if they do with the focus on the acronym, the student never remember the caveats
I dunno. It’s over 30 years on and I remember, and, like I said, I don’t think we were even taught the acronym. (I’m over 95% sure we weren’t.) Of course, it was the Accelerated Algebra class, so maybe that has something to do with it, as we were on the honors track and were just expected to learn it.
IMO that’s really key.
The smart kids in the smart classes are taught different info differently from the ordinary kids in the ordinary classes. And then the smart kids take away different learnings than the ordinary kids.
Now compare teaching in a well-funded district where most parents are college-educated with a badly-funded district out in the country or in the inner city. Vast differences there too.