Post #4436504

@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.