Ivory: Ratioed

Not all of the values we want Ivory to represent can be expressed as a sum-of-products: some require a ratio of two sums-of-products, which we’ll write with a slash /{scheme}, like (/ a b). This structure will always contain two sum-of-products expressions: we’ll call the first the numerator (which is a, in that example) and the second the denominator (b in that example). The denominator should never be (+) (Ivory’s representation of zero).

Ratios are a powerful numerical representation, but they introduce some redundancies which must be normalised.

Division by one

A ratio with denominator (×) (which is Ivory’s representation of the number one) is equivalent to its numerator. Hence the normal form for all Ivory numbers will be a ratio-of-sums-of-products: if a number is not already such a ratio, we will normalise it by making it the numerator of a ratio, with a denominator of (+ (×)).

Common factors

If a particular value appears in every product in both the numerator and denominator of a ratio, then it is a “common factor” which does not affect the result and can hence be removed. Symbolic factors can be found by simply checking for their presence or absence in the elements of a product. Since we represent natural numbers in place-value form, an explicit factorisation algorithm must be used.

Rationalising denominators

Ratios involving symbolic values can be represented in multiple ways: some with symbolic values in the numerator, some with them in the denominator, and some with both. For example, a negative ratio may have a factor of negative one in either the numerator or denominator. We will avoid this ambiguity by choosing the form which has the fewest symbolic values in the denominator. This is called rationalising the denominator when applied to fractions involving radicals. It relies on the following laws:

• A ratio (/ n n) is equal to (×), i.e. to the number one (whenever n is non-zero).
• (×) is the identity value for multiplication, i.e. multiplying by (×) leaves any number unchanged.
• Hence multiplication by (/ n n) leaves any number unchanged (for non-zero n)

This fact can be used to introduce extra factors into the numerator and denominator of a ratio. We can eliminate symbolic factors from the denominator through an appropriate choice of n. For example, given a ratio like (/ 3 i) (where (= (× i i) (- (×)))), we can eliminate i from the denominator by multiplying the ratio by (/ i i):

(= (/ 3 i)
   ; Multiply by (/ i i) to eliminate i from the denominator
   (× (/ 3 i) (/ i i))
   (/ (× 3 i) (× i i))
   (/ (× 3 i) (- (×)))
   ; Multiply by (/ (- (×)) (- (×))) to eliminate (- (×)) from the denominator
   (× (/ (× 3 i) (- (×))) (/ (- (×)) (- (×))))
   (/ (× 3 i (- (×))) (× (- (×)) (- (×))))
   (/ (× 3 i (- (×))) (×)))