## nullary

posted on 11 Nov 2019

Nullary is a base-less numeric encoding. Unlike unary, which requires `O(n)` symbols to encode the number `n`, nullary encoding still requires just `O(log(n))` symbols.

Nullary is the foundation of real numerology.

Nullary is a variant of the ‘xenotation’ described by Nick Land in his essays The Tic Xenotation and TX2.

### why not use the xenotation

Tic Xenotation has an unnecessary base case.

``````Xenotation
----------

: -> 2
xy -> x * y
(x) -> the x'th prime
``````

Nullary fixes this with a natural base case.

``````Nullary
-------
-> 1
xy -> x * y
(x) -> the x'th prime
``````

### canonical form

The expressions `()(())` and `(())()` encode the same value. The number of ways to encode a value is a key property of numbers. Nevertheless, it is useful to have a canonical encoding. The canonical encoding is defined to be the one which places the largest factors first. This means `(())()` is the canonical form of `()(())`. Another way to express this is to say that the canonical form is the first item in a lexical sort if `'(' < ')'`.

### forecast

Mr Land mentions an extension to identify 0 and 1:

``````((-P)): -> 0
(-P): -> 1
: -> 2
``````

The idea is that ‘`(-P)`’ is an operator that unwraps an expression. In this case, it is applied to ‘`:`’. This formulation is unsatisfactory for a number of reasons…

For now, work some examples to get familiar with nullary. How do you encode your favorite number?

`(()())(())()`