# Ivory:: Negatives And Inverses

Negatives can be represented using products, if we adjoin -1.

Inverses can be lifted to the top level: one big fraction (rational expression), a/b, where a and b are sums of products (polynomials). We can probably allow fractional powers of naturals too, but not of arbitrary expressions.

Having each expression be a fraction lets us avoid special symbols for each inverse value.

Probably best to have ^ be a function, which spits out appropriate expressions; rather than a constructor appearing in the expression format. What powers do we want? - (^ n m) for arbitrary n and natural m can be expanded into a product, e.g. (^ n 5) = (/ (× n n n n n) 1) - (^ n -1) for arbitrary n is an inverse, just flip its fraction, e.g. (^ (/ a b) -1) = (/ b a) - (^ n (× -1 m)) for arbitrary n and natural m: expand out and invert, e.g. (^ (/ a b) (× -1 3)) = (× (/ b a) (/ b a) (/ b a)) - (^ n m) for arbitrary n and integer m, based on the above cases - (^ n m) for rational n and rational m: take inverse if needed, take root if needed, expand out if needed. I think that’s a valid algorithm? Maybe property-check it against a real rational implementation.

PARSING - Parse unicode COMBINING OVERLINE - Allow negative
indeterminates, GA units, etc. so long as they’re completely overlined
(i.e. `d₀`

needs a combining character for the `d`

and another for the `₀`

) - Allow partially-negative literals
- Interpret these as negative digits - Can we just parse by peeking for
a COMBINING OVERLINE and, if coming up, consuming everything until we
hit a char without a subsequent COMBINING OVERLINE? - Recursively parse
every other character, and negate the result - Would need special-case
to handle negative digits - Ambiguity for reciprocal of a negative,
e.g. `3̅²̅`

could mean `(= (- (expt 3 2)) 9̅)`

or
`(= (expt 3̅ 2̅) 1/9)`

- The former would be the result of a
simple every-other-character parser - The latter can be represented with
`3̅⁻²`