
10.4: Be aware of a Calculator widget bug
It is nonsense to suggest that the implied order of operations "cannot be right or wrong"  it is right, in the sense that failing to adhere to it will give you the wrong results.
People writing equations are always going to follow the rules, because there are no other alternative "conventions". If there is only one "convention", and it must be followed for accuracy, then it is not a convention at all, but a rule. If you receive the equation a + b * c, then the correct formula is a + (b * c) because the person writing the equation would have written (a + b) * c if they had meant it to be computed that way. They are not going to think "well, under a different convention, a + b * c does equal (a + b) * c, so its OK to leave it unparenthesised when what I mean is (a + b) * c". You said it yourself: "there are often competing conventions for things until a field is welldeveloped enough that a standard emerges" (my emphasis). The field of arithmetic is indeed well developed, and the implied order of operator precedence is indeed a standard. And, for reference, scientific and graphical calculators, along with programming languages, did not adopt any mathematical "conventions". They adopted a rule which was already (long, long already) a standard, and they had no choice but to do that. The notion that they arbitrarily chose from amongst a selection of possible implied orders of operation is just plain wrong. As for the calculator widget, I would not call its behaviour a bug  because the behaviour is clearly intended, and it is correctly recreating the behaviour of the device it is emulating. A calculator that can only handle dualoperand operations isn't buggy or wrong (it correctly calculates those dualoperand operations), just featurepoor.
10.4: Be aware of a Calculator widget bug
I am most of the way through a 2nd year university course that deals with defining and constructing numbers, addition, multiplication etc. Our PEMDAS convention could have easily been different (say PEASDM), in which case you would get 1+2*3=9.
10.4: Be aware of a Calculator widget bug
quote  "Whoever built the first pocket calculator established a new convention for inputting equations, which is sometimes more useful, and often less useful. This convention was established presumably because calculators originally could not handle PEMDAS" 
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