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 $d_3d_2d_1d_0$ in base b is given by the polynomial form
$d_3\times b^3+d_2\times b^2+d_1\times b+d_0$.
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
$1\times16^3+15\times16^2+3\times16+10$.
Upon 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 sexagesimal Babylonian 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.

Bijective numeration systems

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.

Base one (unary numeral system)

Main article: Unary numeral system

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 $d_3d_2d_1d_0$ given by the polynomial form can be simplified into $d_3+d_2+d_1+d_0$ since $b^n=1$ for all n. Non-standard features of this system include:

• The value of a digit does not depend on its position. Thus, one can easily argue that unary is not a positional system at all.
• Introducing a radix point in this system will not enable representation of non-integer values.
• The single numeral represents the value 1, not the value 0=b-1.
• The value 0 cannot be represented (or is implicitly represented by an empty digit string).

Signed-digit representation

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).

Gray code

Main article: Gray code

The reflected binary code, also known as the Gray code, is closely related to binary numbers, but some bits are inverted, depending on the values of the higher order bits.

Bases that are not positive integers

A few positional systems have been suggested in which the base b is not a positive integer.

Negative base

Main article: negative base

Negative-base systems include negabinary, negaternary and negadecimal; in base −b the number of different numerals used is b. All integers, positive and negative, can be represented without a sign.

Complex base

Main article: Quater-imaginary base

In purely imaginary base bi the b2 numbers from 0 to b2 − 1 are used as digits.
It can be generalized on other complex bases: Complex base systems.

Non-integer base

In these systems, the number of different numerals used clearly cannot be b. Example: Golden ratio base (phinary).