8/2(2+2) what do you come up with?

[fishwax]

93. but it is arbitrary, as your own final few sentences show

As I said, these conventions have shifted over the years. That's because they're arbitrary, and the assumptions underscoring the conventions have shifted. Nothing about math has shifted--the computation itself isn't arbitrary. But how we organize and present it as a language is/has. In this case these conventions have shifted, as I noted in my original post (and as you reinforce here) because of the increasing prominence of programmed calculations, among other things. You can't really put the equation, as written, in most of the calculators or computers that kids would use for calculation, because doing so would require you to insert the operation symbol between two juxtaposed values after the slash. (Or, the program might be written to enter it automatically.) Again, that's not because this is the one and only correct way to do it (as, for instance, the non-arbitrary and universally true fact adding 2 to 2 gives you 4) ... rather, it's because that is now the governing convention in that context. (Indeed, it is the far more common convention now, precisely because of the prominence of computers as computational tools.)

This fact that this equation uses all numerals and no variables makes it particularly useful for revealing (and sowing, lol) this sort of distinction/argument. I think the ambiguity would be quite a bit clearer if the equation were written in variables. Now, everybody can agree that bc = b * c, right? But it is also absolutely true that bc = (b * c). So if you're given the expression a/bc, you've got an ambiguity problem, because it could be a/b * c or it could be a/(b * c).