Post #4436506

2026-08-07 13:55 UTC

@FishFace@ioc.exchange @IceWolf@masto.brightfur.net Alright, let's see where this idea leads. We're sticking with the rules of algebra here and adopting the convention that division is right associative: a ÷ b ÷ c = a ÷ (b ÷ c) = a ÷ b x c We now have an algebra that is logically inconsistent. The convention of left associativity was not chosen arbitrarily. It is a necessity in the algebraic system as it's defined. I will grant you that my phrasing "a fundamental property of division" was misleading. What I should have said is that once you adopt the notation "numerator ÷ denominator" to express division, the rest follows as a logical consequence. The lessons we received on order of operations were not given to us to resolve an ambiguity, they were mnemonics to help us carry out the operations correctly. I'm not going to engage with examples outside of algebra such as RPN or calculators because those are completely different systems with different rules and goals. Once we step outside the rules of algebra, yes, I grant you, there are many ways to express math and computation.

Replies (1)

@FishFace@ioc.exchange 2026-08-07 14:11

@glynwolf@tiggi.es @IceWolf@masto.brightfur.net > a ÷ b ÷ c = a ÷ (b ÷ c) = a ÷ b x c Can you justify the final equality? I would write: a ÷ b ÷ c = a ÷ (b ÷ c) = (a ÷ b) × c = (a × c) ÷ b and if you like I'll justify those latter equalities using basic properties of arithmetical operations. Having declared that division is right-associative, but not declared how we will handle expressions with mixed multiplication and division, we still have two choices of how to interpret a ÷ b × c: is it (a ÷ b) × c or a ÷ (b × c)? This is why I don't think you can justify your final equality without making a further notational convention. If you adopt either of the conventions, "do division before multiplication" or "do mixed multiplication and division left-to-right" you will get your equality but no inconsistency. If you adopt either of the conventions "do multiplication before division" or "do mixed multiplication and division right-to-left" you will get the *other* equality, and again no inconsistency. Note that, there IS an inconsistent option here: you cannot both have right-associative division AND do all divisions and multiplications from left to right, for the former necessitates that expressions consisting of multiple division operations be evaluated right-to-left. > What I should have said is that once you adopt the notation "numerator ÷ denominator" to express division, the rest follows as a logical consequence. I still don't agree with this statement. A different way of looking at this is that that notation tells us what to do with expressions involving two quantities only. You still need to adopt a notational convention on what to do with successive operations, unless you make it explicit with brackets. I do think those examples are relevant, because for example, if any inconsistency flowed directly from adopting right-associativity for division written as "a ÷ b", there would be an inconsistency in APL, which use that notation (yes, with an obelus, not a slash) and right-associativity.

Open ##4436504