Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers 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 system. It was used in some early computers[1] and has also been used to solve balance puzzles.[2]
Different sources use different glyphs to represent the three digits in balanced ternary. In this article, T (which resembles a 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 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: "1".[1]
Balanced ternary makes an early appearance in Michael Stifel's book Arithmetica Integra (1544).[3] 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, Augustin-Louis Cauchy, and possibly even the ancient Indian Vedas.[2]
^ abHayes, Brian (2001), "Third base" (PDF), American Scientist, 89 (6): 490–494, doi:10.1511/2001.40.3268. Reprinted in Hayes, Brian (2008), Group Theory in the Bedroom, and Other Mathematical Diversions, Farrar, Straus and Giroux, pp. 179–200, ISBN 9781429938570
^Stifel, Michael (1544), Arithmetica integra (in Latin), apud Iohan Petreium, p. 38.
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