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 comply with the following description of standard positional systems:

In a standard positional numeral system, the base b is a positive integer, and b different numerals are used to represent all non-negative integers. Each numeral represents one of the values 0, 1, 2, etc., up to b-1, but the value also depends on the position of the digit in a number. The value of a digit string like in base b is given by the polynomial form

.

The numbers written in superscript represent the powers of the base used.

For instance, in hexadecimal (b=16), using A=10, B=11 etc., the digit string 1F3A means

.

Introducing a radix point "." and a minus sign "–", all real numbers can be represented.

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.

Certain historical numeral systems like the Babylonian (standard) sexagesimal notation or the Chinese rod numerals could be classified as standard systems of base 60 and 10, respectively (unconventionally counting the space representing zero as a numeral). However, they could also be classified as non-standard systems (more specifically, mixed-base systems with unary components), if the primitive repeated glyphs making up the numerals are considered.