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 radical 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 radical is OK; roots of products seem fine. Roots of sums don’t easily rewrite (not sure of a general normal form for those…)
  - 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 radical requires factorising. That’s slow, but also useful for cancelling common factors in a rational.
How to read and write powers of sums? Would like to avoid parentheses…
- Don’t!
Maybe we can construct roots based on an “index”, e.g. we have (i n) to refer to the nth symbol that squares to -1; how about (√ 5 3) for the cube root of 5? Putting the “base” as first argument lets us default the second to 2, so (√ 5) is short for (√ 5 2).
- Note that (√ n 1) reduces to n
- Composite numbers reduce to products, e.g. (√ 18) = (× (√ 2) (√ 3) (√ 3))
- Products of roots reduce as appropriate, e.g. (× (√ 3) (√ 3)) = 3
The nice thing about this representation, is that we’re not having to deal with all algebraic numbers; we’re just simplifying the symbols that appear in a given expression.

NOTE: The set of algebraic numbers is larger than the set of radicals. Algebraic numbers are roots of arbitrary univariate polynomials, but quintics and above allow roots that can’t be expressed using radicals!