Non-standard positional numeral systems

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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:

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. The standard set of numerals contains the b values 0, 1, 2, etc., up to b − 1, but the value is weighted according to the position of the digit in a number. The value of a digit string like pqrs in base b is given by the polynomial form

${\displaystyle p\times b^{3}+q\times b^{2}+r\times b+s}$.

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

For instance, in hexadecimal (b = 16), using the numerals A for 10, B for 11 etc., the digit string 7A3F means

${\displaystyle 7\times 16^{3}+10\times 16^{2}+3\times 16+15}$,

which written in our normal decimal notation is 31295.

Upon introducing a radix point "." and a minus sign "−", real numbers can be represented up to arbitrary accuracy.

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 were devised by mathematicians or engineers for special academic or technical use.