Post #4436507

2026-08-07 08:35 UTC

@glynwolf@tiggi.es @IceWolf@masto.brightfur.net but left associativity is itself a notational convention. It's not a fundamental property of division: you can't prove that the division operation has this property, you have to decide you're going to follow the convention, and could choose not to. Furthermore, left associativity does not help you in a mixed expression like 12÷3×4. The author is using a nonstandard definition of left associativity and if you look it up on Wikipedia or in a textbook you will see it gives you that a÷b÷c = (a÷b)÷c, but doesn't say what to do when you have a multiplication instead of a second division. In our schooling, left associativity of division is a consequence of our order of operations conventions which say directly to do multiplication and division in order from left to right. If you do this with a÷b÷c you can see you immediately get (a÷b)÷c! Examples where this order is not chosen include: * Reverse polish notation * Calculators which evaluate strictly left to right (used to be the only way calculators worked) * Strange programming languages like APL * Dutch classrooms until they agreed to use the more common order (they used to do multiplication before division) If left associativity were an inherent property of division, none of these would actually be capable of doing division, yet somehow they are.

Replies (1)

@glynwolf@tiggi.es 2026-08-07 13:55

@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.

Open ##4436506