3 ms·
The (principle) cube root of the integer -1 is -1, but the (principle) cube root of the complex number -1 is (1 + sqrt 3) / 2. They sure seem like different obj
by dataflow 2y ago
The (principle) cube root of the integer -1 is -1, but the (principle) cube root of the complex number -1 is (1 + sqrt 3) / 2. They sure seem like different objects.
Though I guess whether you want to blame that on the operators vs. the objects themselves can be left to your taste, but I'm not sure what "these objects are equivalent" would mean if the behaviors are left unspecified as characteristics of the objects (which would be the former case).
- auggierose 2y agoThat is indeed a nice feature of types: You can overload notions. In this case you overload the notion "principal cube root" (PCR) to mean different things for ℤ and for ℂ. But really, "principal cube root" as you would like it to behave is not well-defined just for a number, you also need to provide the algebraic structure you consider it in, as in PCR(ℤ, -1) = -1, and PCR(ℂ, -1) = (1 + sqrt 3) / 2. Alternatively, just set PCR(-1) = (1 + sqrt 3) / 2. That makes probably the most sense, as there is not much value in a PCR notion for integers in the first place.
- lupire 2y ago"Principal" root isn't an interesting mathematical object. It's just an arbitrary way to choose a preferred element from an equivalence class, when you're too lazy to consider the whole class and you want to pretend that 'implies" ("only if") is is the same as " biconditional" ('if and only if").
- dataflow 2y ago> when you're too lazy to consider the whole class Well then consider the whole class? The whole classes are different too, I don't get your point.