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A ternary /ˈtɜːrnəri/ numeral system (also called base 3) has three as its base. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit is equivalent to log_{2} 3 (about 1.58496) bits of information.
Although ternary most often refers to a system in which the three digits are all non–negative numbers; specifically 0, 1, and 2, the adjective also lends its name to the balanced ternary system; comprising the digits −1, 0 and +1, used in comparison logic and ternary computers.
Comparison to other bases
×  1  2  10  11  12  20  21  22  100 
1  1  2  10  11  12  20  21  22  100 
2  2  11  20  22  101  110  112  121  200 
10  10  20  100  110  120  200  210  220  1000 
11  11  22  110  121  202  220  1001  1012  1100 
12  12  101  120  202  221  1010  1022  1111  1200 
20  20  110  200  220  1010  1100  1120  1210  2000 
21  21  112  210  1001  1022  1120  1211  2002  2100 
22  22  121  220  1012  1111  1210  2002  2101  2200 
100  100  200  1000  1100  1200  2000  2100  2200  10000 
Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal 365 or senary 1405 corresponds to binary 101101101 (nine digits) and to ternary 111112 (six digits). However, they are still far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary and septemvigesimal.
Ternary  1  2  10  11  12  20  21  22  100 

Binary  1  10  11  100  101  110  111  1000  1001 
Senary  1  2  3  4  5  10  11  12  13 
Decimal  1  2  3  4  5  6  7  8  9 
Ternary  101  102  110  111  112  120  121  122  200 
Binary  1010  1011  1100  1101  1110  1111  10000  10001  10010 
Senary  14  15  20  21  22  23  24  25  30 
Decimal  10  11  12  13  14  15  16  17  18 
Ternary  201  202  210  211  212  220  221  222  1000 
Binary  10011  10100  10101  10110  10111  11000  11001  11010  11011 
Senary  31  32  33  34  35  40  41  42  43 
Decimal  19  20  21  22  23  24  25  26  27 
Ternary  1  10  100  1000  10000 

Binary  1  11  1001  11011  1010001 
Senary  1  3  13  43  213 
Decimal  1  3  9  27  81 
Power  3^{0}  3^{1}  3^{2}  3^{3}  3^{4} 
Ternary  100000  1000000  10000000  100000000  1000000000 
Binary  11110011  1011011001  100010001011  1100110100001  100110011100011 
Senary  1043  3213  14043  50213  231043 
Decimal  243  729  2187  6561  19683 
Power  3^{5}  3^{6}  3^{7}  3^{8}  3^{9} 
As for rational numbers, ternary offers a convenient way to represent 1/3 as same as senary (as opposed to its cumbersome representation as an infinite string of recurring digits in decimal); but a major drawback is that, in turn, ternary does not offer a finite representation for 1/2 (nor for 1/4, 1/8, etc.), because 2 is not a prime factor of the base; as with base two, onetenth (decimal1/10, senary 1/14) is not representable exactly (that would need e.g. decimal); nor is onesixth (senary 1/10, decimal 1/6).
Fraction  1/2  1/3  1/4  1/5  1/6  1/7  1/8  1/9  1/10  1/11  1/12  1/13 

Ternary  0.1  0.1  0.02  0.0121  0.01  0.010212  0.01  0.01  0.0022  0.00211  0.002  0.002 
Binary  0.1  0.01  0.01  0.0011  0.001  0.001  0.001  0.000111  0.00011  0.0001011101  0.0001  0.000100111011 
Senary  0.3  0.2  0.13  0.1  0.1  0.05  0.043  0.04  0.03  0.0313452421  0.03  0.024340531215 
Decimal  0.5  0.3  0.25  0.2  0.16  0.142857  0.125  0.1  0.1  0.09  0.083  0.076923 
Sum of the digits in ternary as opposed to binary
The value of a binary number with n bits that are all 1 is 2^{n} − 1.
Similarly, for a number N(b, d) with base b and d digits, all of which are the maximal digit value b − 1, we can write:
 N(b, d) = (b − 1)b^{d−1} + (b − 1)b^{d−2} + … + (b − 1)b^{1} + (b − 1)b^{0},
 N(b, d) = (b − 1)(b^{d−1} + b^{d−2} + … + b^{1} + 1),
 N(b, d) = (b − 1)M.
 bM = b^{d} + b^{d−1} + … + b^{2} + b^{1} and
 −M = −b^{d−1} − b^{d−2} − … − b^{1} − 1, so
 bM − M = b^{d} − 1, or
 M = b^{d} − 1/b − 1.
Then
 N(b, d) = (b − 1)M,
 N(b, d) = (b − 1)(b^{d} − 1)/b − 1,
 N(b, d) = b^{d} − 1.
For a threedigit ternary number, N(3, 3) = 3^{3} − 1 = 26 = 2 × 3^{2} + 2 × 3^{1} + 2 × 3^{0} = 18 + 6 + 2.
Compact ternary representation: base 9 and 27
Nonary (base 9, each digit is two ternary digits) or septemvigesimal (base 27, each digit is three ternary digits) can be used for compact representation of ternary, similar to how octal and hexadecimal systems are used in place of binary.
Practical usage
In certain analog logic, the state of the circuit is often expressed ternary. This is most commonly seen in CMOS circuits, and also in transistor–transistor logic with totempole output. The output is said to either be low (grounded), high, or open (highZ). In this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a certain reference, or at a certain voltage level, the state is said to be high impedance because it is open and serves its own reference. Thus, the actual voltage level is sometimes unpredictable.
A rare "ternary point" in common use is for defensive statistics in American baseball (usually just for pitchers), to denote fractional parts of an inning. Since the team on offense is allowed three outs, each out is considered one third of a defensive inning and is denoted as .1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus achieving 2 outs in the 7th inning, his innings pitched column for that game would be listed as 3.2, the equivalent of 3 ^{2}⁄_{3} (which is sometimes used as in alternative by some record keepers). In this usage, only the fractional part of the number is written in ternary form.^{[1]}^{[2]}
Ternary numbers can be used to convey self–similar structures like the Sierpinski triangle or the Cantor set conveniently. Additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have a ternary expression that does not contain any instance of the digit 1.^{[3]}^{[4]} Any terminating expansion in the ternary system is equivalent to the expression that is identical up to the term preceding the last nonzero term followed by the term one less than the last nonzero term of the first expression, followed by an infinite tail of twos. For example: 0.1020 is equivalent to 0.1012222... because the expansions are the same until the "two" of the first expression, the two was decremented in the second expansion, and trailing zeros were replaced with trailing twos in the second expression.
Ternary is the integer base with the lowest radix economy, followed closely by binary and quaternary. This is due to its proximity to e. It has been used for some computing systems because of this efficiency. It is also used to represent threeoption trees, such as phone menu systems, which allow a simple path to any branch.
A form of redundant binary representation called a binary signeddigit number system, a form of signeddigit representation, is sometimes used in lowlevel software and hardware to accomplish fast addition of integers because it can eliminate carries.^{[5]}
Binarycoded ternary
Simulation of ternary computers using binary computers, or interfacing between ternary and binary computers, can involve use of binarycoded ternary (BCT) numbers, with two bits used to encode each trit.^{[6]}^{[7]} BCT encoding is analogous to binarycoded decimal (BCD) encoding. If the trit values 0, 1 and 2 are encoded 00, 01 and 10, conversion in either direction between binarycoded ternary and binary can be done in logarithmic time.^{[8]} A library of C code supporting BCT arithmetic is available.^{[9]}
Tryte
Some ternary computers such as the Setun defined a tryte to be six trits^{[10]} or approximately 9.5 bits (holding more information than the de facto binary byte).^{[11]}
See also
References
 ^ Ashley MacLennan (20190109). "A complete beginner's guide to baseball stats: Pitching statistics, and what they mean". Bless You Boys. Retrieved 20200730.
 ^ "Stats  Team  Pitching". MLB (Major League Baseball). Retrieved 20200730.
 ^ Soltanifar, Mohsen (2006). "On A sequence of cantor Fractals". Rose Hulman Undergraduate Mathematics Journal. 7 (1). Paper 9.
 ^ Soltanifar, Mohsen (2006). "A Different Description of A Family of Middle–α Cantor Sets". American Journal of Undergraduate Research. 5 (2): 9–12.
 ^ Phatak, D. S.; Koren, I. (1994). "Hybrid signed–digit number systems: a unified framework for redundant number representations with bounded carry propagation chains" (PDF). IEEE Transactions on Computers. 43 (8): 880–891. CiteSeerX 10.1.1.352.6407. doi:10.1109/12.295850.
 ^ Frieder, Gideon; Luk, Clement (February 1975). "Algorithms for Binary Coded Balanced and Ordinary Ternary Operations". IEEE Transactions on Computers. C24 (2): 212–215. doi:10.1109/TC.1975.224188.
 ^ Parhami, Behrooz; McKeown, Michael (20131103). "Arithmetic with BinaryEncoded Balanced Ternary Numbers". Proceedings 2013 Asilomar Conference on Signals, Systems and Computers. Pacific Grove, CA, USA: 1130–1133. doi:10.1109/ACSSC.2013.6810470. ISBN 9781479923908.
 ^ Jones, Douglas W. (June 2016). "Binary Coded Ternary and its Inverse".
 ^ Jones, Douglas W. (20151229). "Ternary Data Types for C Programmers".
 ^ Impagliazzo, John; Proydakov, Eduard (20110906). Perspectives on Soviet and Russian Computing: First IFIP WG 9.7 Conference, SoRuCom 2006, Petrozavodsk, Russia, July 3—7, 2006, Revised Selected Papers. Springer. ISBN 9783642228162.
 ^ Brousentsov, N. P.; Maslov, S. P.; Ramil Alvarez, J.; Zhogolev, E. A. "Development of ternary computers at Moscow State University". Retrieved 20100120.
Further reading
 Hayes, Brian (November–December 2001). "Third base" (PDF). American Scientist. Sigma Xi, the Scientific Research Society. 89 (6): 490–494. doi:10.1511/2001.40.3268. Archived (PDF) from the original on 20191030. Retrieved 20200412.
External links
 Ternary Arithmetic
 The ternary calculating machine of Thomas Fowler
 Ternary Base Conversion – includes fractional part, from Maths Is Fun
 Gideon Frieder's replacement ternary numeral system