Ivory: Radicals

To be radical is to grasp things by the root — Karl Marx, Critique of Hegel’s Philosophy of Right

TODO

(expt x y) is tricky:
- When y is natural we get the same type as x (assuming it’s closed under multiplication, which all of our types are)
- When y is integer we get rational numbers or below
- When y is rational we get algebraic and below
- Probably don’t want anything more general than rational, since their semantics involve logarithms, which I don’t know a normal form for
- Unclear what’s the most general x we can support, when y is rational:
  - I think algebraic is OK; roots of products seem fine, roots of sums don’t easily rewrite, which I hope means they’re normalised, right?
  - Roots become more, ahem, complex when dealing with complex, and other hypercomplex numbers; since sums can cancel-out when squared.
  - Seems reasonable to return complex when rooting a negative
  - If there’s a symbolic normal form for other geometric numbers it would be nice to use that
  - Fractional powers of a polynomial should be representable using algebraic-expression, I think?
  - Note: don’t confuse fractional powers of a polynomial with “roots” (substitutions for the indeterminate which evaluate to 0)
Normalising algebraic requires factorising. That’s slow, but is “zero cost” in the sense we’ll never do it unless someone starts taking roots…
Is normalising as simple as factoring, until we get a fixed-point of X = sums-of-products-of-powers-of X?
How to read and write powers of sums? Would like to avoid parentheses…