Editing Positional notation (section)

== Non-standard positional numeral systems ==
{{Main|Non-standard positional numeral systems}}
Interesting properties exist when the base is not fixed or positive and when the digit symbol sets denote negative values. There are many more variations. These systems are of practical and theoretic value to computer scientists.

[[Balanced ternary]]<ref>[[#Knuth|Knuth]], pages 195–213</ref> uses a base of 3 but the digit set is {{mset|{{overline|1}},0,1}} instead of {0,1,2}. The "{{overline|1}}" has an equivalent value of −1. The negation of a number is easily formed by switching the {{overline|&nbsp;&nbsp;}} on the 1s. This system can be used to solve the [[balance problem]], which requires finding a minimal set of known counter-weights to determine an unknown weight. Weights of 1, 3, 9, ..., 3<sup>''n''</sup> known units can be used to determine any unknown weight up to 1 + 3 + ... + 3<sup>''n''</sup> units. A weight can be used on either side of the balance or not at all. Weights used on the balance pan with the unknown weight are designated with {{overline|1}}, with 1 if used on the empty pan, and with 0 if not used. If an unknown weight ''W'' is balanced with 3 (3<sup>1</sup>) on its pan and 1 and 27 (3<sup>0</sup> and 3<sup>3</sup>) on the other, then its weight in decimal is 25 or 10{{overline|1}}1 in balanced base-3.
: {{math|10{{overline|1}}1<sub>3</sub> {{=}} 1 × 3<sup>3</sup> + 0 × 3<sup>2</sup> − 1 × 3<sup>1</sup> + 1 × 3<sup>0</sup> {{=}} 25.}}

The [[factorial number system]] uses a varying radix, giving [[factorial]]s as place values; they are related to [[Chinese remainder theorem]] and [[residue number system]] enumerations. This system effectively enumerates permutations. A derivative of this uses the [[Towers of Hanoi]] puzzle configuration as a counting system. The configuration of the towers can be put into 1-to-1 correspondence with the decimal count of the step at which the configuration occurs and vice versa.

{|class="wikitable" style="text-align:center;" border="1"
!align="left" |Decimal equivalents
|width="6%" |−3
|width="6%" |−2
|width="6%" |−1
|width="6%" |0
|width="6%" |1
|width="6%" |2
|width="6%" |3
|width="6%" |4
|width="6%" |5
|width="8%" |6
|width="8%" |7
|width="8%" |8
|-
!align="left" |Balanced base 3
|{{overline|1}}0
|{{overline|1}}1
|{{overline|1}}
|0
|1
|1{{overline|1}}
|10
|11
|1{{overline|1}}{{overline|1}}
|1{{overline|1}}0
|1{{overline|1}}1
|10{{overline|1}}
|-
!align="left" |Base −2
|1101
|10
|11
|0
|1
|110
|111
|100
|101
|11010
|11011
|11000
|-
!align="left" |Factoroid
| || || ||0 ||10 ||100 ||110 ||200 ||210 ||1000 ||1010 ||1100
|}