Editing Balanced ternary (section)

{{Short description|Numeral system using the values -1, 0 and 1}}
{{numeral systems}}
'''Balanced ternary''' is a [[ternary numeral system]] (i.e. base 3 with three [[Numerical digit|digits]]) that uses a balanced [[signed-digit representation]] of the [[integer]]s in which the digits have the values [[−1]], [[0]], and [[1]]. This stands in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2. 
The balanced ternary system can represent all integers without using a separate [[minus sign]]; the value of the leading non-zero digit of a number has the sign of the number itself. The balanced ternary system is an example of a [[Non-standard positional numeral systems|non-standard positional numeral system]]. It was used in some early computers<ref name=setun/> and has also been used to solve [[balance puzzle]]s.<ref name=hayes/>

Different sources use different glyphs to represent the three digits in balanced ternary. In this article, T (which resembles a [[typographical ligature|ligature]] of the minus sign and 1) represents [[−1]], while [[0]] and [[1]] represent themselves. Other conventions include using '−' and '+' to represent −1 and 1 respectively, or using [[Greek alphabet|Greek letter]] [[theta]] (Θ), which resembles a minus sign in a circle, to represent −1. In publications about the [[Setun]] computer, −1 is represented as overturned 1: "<span class="plainlinks"></span><span style="display:inline-block;vertical-align:-0.05em;transform:matrix(-1, 0, 0, -1, 0, 0);">1</span>".<ref name=setun>{{cite book|title=Programming|year=1963|location=Moscow|author=N. A. Krinitsky |author2=G. A. Mironov |author3=G. D. Frolov|editor=M. R. Shura-Bura|language=ru|chapter=Chapter 10. Program-controlled machine Setun}}</ref>

Balanced ternary makes an early appearance in [[Michael Stifel]]'s book ''Arithmetica Integra'' (1544).<ref>{{citation
 | last = Stifel | first = Michael | author-link = Michael Stifel
 | language = Latin
 | page = 38
 | title = Arithmetica integra
 | url = https://archive.org/stream/bub_gb_ywkW9hDd7IIC#page/n85/mode/2up
 | year = 1544| publisher = apud Iohan Petreium }}.</ref> It also occurs in the works of [[Johannes Kepler]] and [[Léon Lalanne]]. Related signed-digit schemes in other bases have been discussed by [[John Colson]], [[John Leslie (physicist)|John Leslie]], [[Augustin-Louis Cauchy]], and possibly even the ancient Indian [[Vedas]].<ref name=hayes>{{citation | first=Brian|last=Hayes|authorlink=Brian Hayes (scientist)|title=Third base|journal=American Scientist|url=http://bit-player.org/bph-publications/AmSci-2001-11-Hayes-ternary.pdf|year=2001|volume=89|issue=6|pages=490&ndash;494|doi=10.1511/2001.40.3268}}. Reprinted in {{citation|title=Group Theory in the Bedroom, and Other Mathematical Diversions|first=Brian|last=Hayes|authorlink=Brian Hayes (scientist)|publisher=Farrar, Straus and Giroux|year=2008|isbn=9781429938570|pages=179–200|url=https://books.google.com/books?id=1ZkYEFi3DMMC&pg=PA179}}</ref>