Ivory: Manifest Decimation

If you can count to two, you can count to anything! — Constable Cuddy, Men At Arms by Terry Pratchett

Dyadic numbers are binary, floating point and arbitrary-precision. They are numbers of the form (/ a (^ 2 n)) for any integer a and natural n. Note that when n is zero, this simplifies to the integer a:

(= (/ a (^ 2 0))
   (/ a 1)
   a)

Larger values of n give halves, quarters, eights, etc. dyadic is a ring which is closed under plus, times, max, min, etc. Since dyadic contains every integer, but only those fractions whose denominator is a power of 2, it is not closed under division: for example, 1 and 3 are dyadic, but (/ 1 3) is not (since the denominator 3 is not a power of 2).

Ivory includes dyadic since it is particularly simple, similar to the popular IEEE754 format, and is commonly used in efficient numerical algorithms.

Beyond binary

ALL YOUR BASE ARE BELONG TO US.

— CATS, Zero Wing

We can generalise dyadic by considering other bases besides 2. Note that making the base variable, like (/ a (^ b n)), allows any rational to be represented; although raising to a power is redundant in that case, since (/ a b) would suffice. However, there is more structure to expose before we get that far!

Each natural gives rise to a different floating-point number system: besides base 2, other reasonable choices for Ivory include bases 3, 6, 8, 10, 12, 16, 60 and 360. Such number systems can, by definition, represent fractions whose denominator is a power of the base; like powers of 2 for dyadic. This also extends to powers of that base’s factors: for example, 3 is a factor of 60, so base 60 can represent a fraction like (/ 1 (^ 3 7). Finally, products of such powers can also be represented, e.g. both 3 and 4 are factors of 12, so base 12 can represent (/ 1 (× (^ 3 3) (^ 4 5))). Note that 2 is also a factor of 12, so that last example could instead be expressed as (/ 1 (× (^ 3 3) (^ 2 10))).

┌─────────┬─────────┐
│ dozenal │ decimal │
├─ ─ ─ ─ ─┴─ ─ ─ ─ ─┤
│    sexagesimal    │
└───────────────────┘

We require every level in Ivory to be a superset of those above, so any floating-point level below dyadic must use a base with a factor of 2, AKA an an even number. This rules out base 3, base 15, etc.

The question is which other factors to include? The most natural is a factor of 3, giving us base 6; or the equivalent yet more popular base 12. However, either of those would prevent us including a level for base 10, which is the most widespread number system in the world! We reconcile these alternatives by choosing neither: instead, the sexagesimal level provides base 60 fractions; with mezzanine siblings for dozenal (base 12) and decimal (base 10).

Notice that this arrangement is completely unambiguous:

• Fractions whose denominator is a power of 2, like (/ 1 2), (/ 13 64), (/ -9 16), etc. live in dyadic. Hence they are also decimal, dozenal and sexagesimal by inclusion.
• Fractions whose denominator involves powers of 2 and 3, like (/ 1 3), (/ -5 6), etc. live in dozenal and are included in sexagesimal. They are not dyadic or decimal.
• Fractions whose denominator involves powers of 2 and 5, like (/ 1 5), (/ -3 20), etc. live in decimal and are included in sexagesimal. They are not dyadic or dozenal.
• Fractions whose denominator involves powers of 2, 3 and 5, like (/ -2 45), live in sexagesimal. The yare not dyadic, decimal or dozenal.

After these, the next factor to consider would be 7, but such number systems do not see much use. Instead, we proceed on to the full rational level!