Ivory: Numbers In Scheme

    ┌─┐ ┌──┐ ┌─┐
    │ └─┘  └─┘ │
    │ integer  │
    ├──────────┤
    │ rational │
    ├──────────┤
    │   real   │
    ├──────────┤
    │ complex  │
    ├──────────┤
    │  number  │
────┴──────────┴────

Scheme uses the asterisk * for multiplication, since it’s easy to type on a standard US keyboard. I prefer the usual × symbol, for clarity. Hence we’ll define × to be *, so that both symbols will mean multiplication:

(define × *)

Numbers in Scheme can be exact or inexact. Ivory is only concerned with exact numbers, so we’ll ignore the latter. Scheme arranges its types in a “numerical tower”, where each level is a super-set of the ones above, including:

• integer: Whole numbers, positive and negative.
• rational: All fractions, positive and negative.
• real: This supposedly includes the whole “number line”, but is actually rather silly since almost all of the “real numbers” can’t be represented.
• complex: This uses a clever trick to represent square roots of negative numbers. If you’ve encountered it before, great; if not, don’t worry since Ivory does not include this level (we instead define a mezzanine inside the geometric level)
• number: This is the top level, containing every numeric value.

These are cumulative, so an integer like -42 is also a rational (e.g. you can think of it like -42/1), a real and a complex (like -42+0i, if you know what that means).

This basic framework is the inspiration for Ivory.

Numbers in Racket

        ┌─┐ ┌──┐ ┌─┐
        │ └─┘  └─┘ │
        │   zero   │
       ┌┴──────────┴┐
       │  natural   │
      ┌┴────────────┴┐
      │   integer    │
     ┌┴──────────────┴┐
     │    rational    │
     ├────────────────┤
     │      real      │
    ┌┴────────────────┴┐
    │     complex      │
    ├──────────────────┤
    │      number      │
────┴──────────────────┴────

Racket already extends Scheme’s standard numerical tower, and pins-down some details that Scheme leaves open. In particular:

• Scheme allows levels between complex and number, but Racket doesn’t provide any.
• Racket’s foundational number type adds nothing else to complex (they’re just synonyms)
• The only exact numbers in Racket’s real level are rational. Hence, as far as Ivory is concerned, those levels are the same.

Racket does add some “attic levels”, which are strict subsets of integer:

• natural is a sub-set of integer without negatives. It is closed under +, ×, gcd, lcm, max, min, etc.; as well as quotient and remainder excluding the divisor 0, and expt excluding (expt 0 0).
• zero is a sub-set of natural containing only 0. It is closed under +, ×, gcd, lcm, max, min, etc.

Even more fine-grained structure is described in the Typed Racket documentation (e.g. byte is a subset of natural between 0 and 255) but Ivory does not make such size-based distinctions.

Code

This post defines Racket code, including a RackCheck test suite to check the claims we make on this page. Here’s a URI containing all of the generated code:

DOWNLOAD RACKET CODE

raco test: (submod (file "num.rkt") test)
  ✓ property ×-matches-* passed 100 tests.
  ✓ property integer?-implies-rational? passed 100 tests.
  ✓ property rational?-implies-number? passed 100 tests.
  ✓ property natural?-implies-integer? passed 100 tests.
  ✓ property natural-is-closed passed 100 tests.
26 tests passed