4 ms·

A ring [0] might be too general, but I can't think of anything that isn't a ring that I'd want to call a "number" class. A field [1] is less general and might b

A ring [0] might be too general, but I can't think of anything that isn't a ring that I'd want to call a "number" class. A field [1] is less general and might be a better fit. It probably depends whether you want a matrix to be able to be a "number," which I'd be inclined to do. [0] https://en.wikipedia.org/wiki/Ring_(mathematics) https://en.wikipedia.org/wiki/Ring_(mathematics) [1] https://en.wikipedia.org/wiki/Field_(mathematics) https://en.wikipedia.org/wiki/Field_(mathematics)

red_trumpet 6y ago
Natural numbers don't form a ring, and integers don't form a field.
- 6gvONxR4sf7o 6y ago
  Good callout. What would you use if you had to restrict operator overloading to an appropriate “number” abstraction?
klyrs 6y ago
The type systems underlying computer algebra systems like Sage, GAP, etc. are immensely more complex than this. Read the source of Sage (lots of python/cython) and the source of GAP (pure c; everything is basically a void *). Operator overloading is immensely helpful for entire classes of objects -- groups, rings, fields, categories, combinatorial objects... some of them are kinda number-like, but for example groups can support multiplication and division but not addition/subtraction (unless it's an abelian group and you'd rather represent it that way). I've been writing C for nearly 30 years, and GAP is a slog to read. Sage makes this stuff way easier. In short... math isn't just arithmetic, isn't even arithmetic of complex tensors. Math is way bigger than most programmers seem to comprehend. I'm on the other side of the DK curve; I know that I've only scratched the surface despite a decade of dedicated study
dllthomas 6y ago
If we're actually requiring things be a ring, rather than approximating something that is a ring, that rules out floating point numbers (lack of associativity, at least).

red_trumpet 6y ago

Natural numbers don't form a ring, and integers don't form a field.

6gvONxR4sf7o 6y ago
Good callout. What would you use if you had to restrict operator overloading to an appropriate “number” abstraction?

klyrs 6y ago

The type systems underlying computer algebra systems like Sage, GAP, etc. are immensely more complex than this. Read the source of Sage (lots of python/cython) and the source of GAP (pure c; everything is basically a void *). Operator overloading is immensely helpful for entire classes of objects -- groups, rings, fields, categories, combinatorial objects... some of them are kinda number-like, but for example groups can support multiplication and division but not addition/subtraction (unless it's an abelian group and you'd rather represent it that way). I've been writing C for nearly 30 years, and GAP is a slog to read. Sage makes this stuff way easier. In short... math isn't just arithmetic, isn't even arithmetic of complex tensors. Math is way bigger than most programmers seem to comprehend. I'm on the other side of the DK curve; I know that I've only scratched the surface despite a decade of dedicated study

dllthomas 6y ago

If we're actually requiring things be a ring, rather than approximating something that is a ring, that rules out floating point numbers (lack of associativity, at least).