Non-standard positional numeral systems
Non-standard positional numeral systems here designates numeral systems that may loosely be described as positional systems, but that do not entirely comply with the following description of standard positional systems:
This article summarizes facts on some non-standard positional numeral systems. In most cases, the polynomial form in the description of standard systems still applies.
Some historical numeral systems may be described as non-standard positional numeral systems. E.g., the sexagesimal Babylonian notation and the Chinese rod numerals, which can be classified as standard systems of base 60 and 10, respectively, counting the space representing zero as a numeral, can also be classified as non-standard systems, more specifically, mixed-base systems with unary components, considering the primitive repeated glyphs making up the numerals.
However, most of the non-standard systems listed below have never been intended for general use, but are deviced by mathematicians or engineers for special academic or technical use.
A bijective numeral system with base b uses b different numerals to represent all non-negative integers. However, the numerals have values 1, 2, 3, etc. up to and including b, whereas zero is represented by an empty digit string. For example, it is possible to have decimal without a zero.
Unary is the bijective numeral system with base b = 1. In unary, one numeral is used to represent all positive integers. The value of the digit string pqrs given by the polynomial form can be simplified into p + q + r + s since bn = 1 for all n. Non-standard features of this system include:
In some systems, while the base is a positive integer, negative digits are allowed. Non-adjacent form is a particular system where the base is b = 2. In the balanced ternary system, the base is b = 3, and the numerals have the values −1, 0 and +1 (rather than 0, 1 and 2 as in the standard ternary system, or 1, 2 and 3 as in the bijective ternary system).
...
Wikipedia