# A proper comprehensive rule for arithmetic operation precendence

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## Summary:

The internet is alive with a "math problem" that is really a question of what should be the proper precedence for arithmetic operations when parentheses are not used to unambiguously denote precedence
Here is a "problem" that is "befudding" folks on the internet. To the less mathematically mature, it might be vexing, but to us mathematically mature, we should have a clear comprehensive meme so that stupid things like this don't happen.

My opinion is as follows:

The baseline unambiguous notation is to have every operation have arguments that are expressions that are encapsulated inside parentheses. A function call supplies its own parentheses, so that encapsulation is done automatically (I'll leave out the situations dealing with syntactical sugar of doing "ln x", etc.). It also can be presumed that any division notated by a vinculum de facto encapsulates both the numerator & denominator, and with the result being itself encapsulated. Similarly, an exponentiation can always be presumed to be self-encapsulating, outside of a parenthesization between the base & exponent. From here, it can also be presumed that a multiplication that is denoted via adjacent-placement takes precendece over addition/subtraction (and would be of no consequence with multiplication, as that is commutative). With these rules in place, the standard interpretation of polynomial notation can be unambiguous (i.e., a polynomial only has addition/subtraction, mulitplication & exponentation, so division does not need to be considered).

Outside of this unambiguous schema, it can be considered ambiguous whether a declared arithmetic sign is meant to be interpreted as having everything on the LHS being evaluated first, and then only the first encapsulated expression being used on the RHS, or if it is to have the common computer-language rule of multiplication & division collectively at higher precedence than addition & subtraction (and with both of each pair at equal precedence with the other). And for the case of adjacent-placement, it could be considered the equivalent of a declared multiplication sign (i.e., saltire, interpunct, asterisk) or a multiplication that has a de facto encapsulation in a similar way that division per vinculum does.

For these reasons, it is good practice to always consider that either of these interpretations could be inferenced, and therefore parentheses should be added so to make the expression be equivalent no matter which interpretation; as a consequence of this any term using adjacent placement multiplication would not have an unencapsulated division - which makes sense since division per vinculum has its de facto encapsulation. In the case of the original problem, it wouldn't have been done like it had, and thus all the electrons wasted in talking about (including these ) would be spared.

I think that teaching this as good structure - in the same way that a language is taught such that anytime a specific construction could be ambiguous (e.g., dangling modifier, so wonderfully presented by my favorite high school teacher by letting his arm dangle while explaining it ), the sentence should be reworded so as to obliterate the ambiguity - would be a good thing in math class.

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all the electrons wasted (including these ) would be spared.
And as we all know, mankind will run out of electrons by 2073, even if we halve our current rate of consumption.