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Aternary/ˈtɜːrnəri/numeral system (also calledbase 3 ortrinary^[1]) hasthree as itsbase. Analogous to abit, a ternarydigit is atrit (trinary digit). One trit is equivalent tolog₂ 3 (about 1.58496) bits ofinformation.

Althoughternary most often refers to a system in which the three digits are all non–negative numbers; specifically0,1, and2, the adjective also lends its name to thebalanced ternary system; comprising the digits−1, 0 and +1, used in comparison logic andternary computers.

Comparison to other bases

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Representations ofinteger numbers in ternary do not get uncomfortably lengthy as quickly as inbinary. For example,decimal 365₍₁₀₎ orsenary1405₍₆₎ corresponds to binary101101101₍₂₎ (ninebits) and to ternary111112₍₃₎ (six digits). However, they are still far less compact than the corresponding representations in bases such asdecimal – see below for a compact way to codify ternary using nonary (base 9) andseptemvigesimal (base 27).

A ternarymultiplication table
×	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	1,000
11	11	22	110	121	202	220	1,001	1,012	1,100
12	12	101	120	202	221	1,010	1,022	1,111	1,200
20	20	110	200	220	1,010	1,100	1,120	1,210	2,000
21	21	112	210	1,001	1,022	1,120	1,211	2,002	2,100
22	22	121	220	1,012	1,111	1,210	2,002	2,101	2,200
100	100	200	1,000	1,100	1,200	2,000	2,100	2,200	10,000

**Numbers from 0 to 3³ − 1 in standard ternary**
Decimal	0	1	2	3	4	5	6	7	8
Ternary	0	1	2	10	11	12	20	21	22
Binary	0	1	10	11	100	101	110	111	1000
Senary	0	1	2	3	4	5	10	11	12

Ternary	100	101	102	110	111	112	120	121	122
Binary	1001	1010	1011	1100	1101	1110	1111	10000	10001
Senary	13	14	15	20	21	22	23	24	25
Decimal	9	10	11	12	13	14	15	16	17

Ternary	200	201	202	210	211	212	220	221	222
Binary	10010	10011	10100	10101	10110	10111	11000	11001	11010
Senary	30	31	32	33	34	35	40	41	42
Decimal	18	19	20	21	22	23	24	25	26

**Powers of 3 in ternary**
Power	3⁰	3¹	3²	3³	3⁴
Ternary	1	10	100	1000	10000
Binary	1	11	1001	11011	1010001
Senary	1	3	13	43	213
Decimal	1	3	9	27	81

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	3⁵	3⁶	3⁷	3⁸	3⁹

As forrational numbers, ternary offers a convenient way to represent⁠1/3⁠ as same as senary (as opposed to its cumbersome representation as an infinite string ofrecurring digits in decimal); but a major drawback is that, in turn, ternary does not offer a finite representation for⁠1/2⁠ (nor for⁠1/4⁠,⁠1/8⁠, etc.), because2 is not aprime factor of the base; as with base two, one-tenth (decimal⁠1/10⁠, senary⁠1/14⁠) is not representable exactly (that would need e.g. decimal); nor is one-sixth (senary⁠1/10⁠, decimal⁠1/6⁠).

**Fractions in ternary**
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
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

Sum of the digits in ternary as opposed to binary

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The value of a binary number withn bits that are all 1 is2ⁿ − 1.

Similarly, for a numberN(b,d) with baseb andd digits, all of which are the maximal digit valueb − 1, we can write:

N(b,d) = (b − 1)b^d−1 + (b − 1)b^d−2 + … + (b − 1)b¹ + (b − 1)b⁰,

N(b,d) = (b − 1)(b^d−1 +b^d−2 + … +b¹ + 1),

N(b,d) = (b − 1)M.

bM =b^d +b^d−1 + … +b² +b¹ and

−M = −b^d−1 − b^d−2 − ... − b¹ − 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 three-digit ternary number,N(3, 3) = 3³ − 1 = 26 = 2 × 3² + 2 × 3¹ + 2 × 3⁰ = 18 + 6 + 2.

Compact ternary representation: base 9 and 27

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Comparison between ternary and nonary
ternary	nonary
00	0
01	1
02	2
10	3
11	4
12	5
20	6
21	7
22	8

Nonary/ˈnɒnəri/ (base 9, each digit is two ternary digits) orseptemvigesimal (base 27, each digit is three ternary digits) can be used for compact representation of ternary, similar to howoctal andhexadecimal systems are used in place ofbinary.

Practical usage

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Use of ternary numbers to balance an unknown integer weight from 1 to 40 kg with weights of 1, 3, 9 and 27 kg (4 ternary digits actually gives 3⁴ = 81 possible combinations: −40 to +40, but only the positive values are useful)

In certain analog logic, the state of the circuit is often expressed ternary. This is most commonly seen inCMOS circuits, and also intransistor–transistor logic withtotem-pole output. The output is said to either be low (grounded), high, or open (high-Z). In this configuration the output of the circuit is actually not connected to anyvoltage 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 highimpedance 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 Americanbaseball (usually just forpitchers), to denote fractional parts of an inning. Since the team on offense is allowed threeouts, 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, hisinnings pitched column for that game would be listed as3.2, the equivalent of3+2⁄3 (which is sometimes used as an alternative by some record keepers). In this usage, only the fractional part of the number is written in ternary form.^[2]^[3]

Ternary numbers can be used to convey self-similar structures like theSierpinski triangle or theCantor 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.^[4]^[5] Any terminating expansion in the ternary system is equivalent to the expression that is identical up to the term preceding the last non-zero term followed by the term one less than the last non-zero 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 lowestradix economy, followed closely bybinary andquaternary. This is due to its proximity to themathematical constant e. It has been used for some computing systems because of this efficiency. It is also used to represent three-optiontrees, such as phone menu systems, which allow a simple path to any branch.

A form ofredundant binary representation called a binary signed-digit number system, a form ofsigned-digit representation, is sometimes used in low-level software and hardware to accomplish fast addition of integers because it can eliminatecarries.^[6]

Binary-coded ternary

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Simulation of ternary computers using binary computers, or interfacing between ternary and binary computers, can involve use of binary-coded ternary (BCT) numbers, with two or three bits used to encode each trit.^[7]^[8] BCT encoding is analogous tobinary-coded decimal (BCD) encoding. If the trit values 0, 1 and 2 are encoded 00, 01 and 10, conversion in either direction between binary-coded ternary and binary can be done inlogarithmic time.^[9] A library ofC code supporting BCT arithmetic is available.^[10]

Tryte

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Someternary computers such as theSetun defined atryte to be six trits^[11] or approximately 9.5bits (holding more information than thede factobinary byte).^[12]

References

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^Kindra, Vladimir; Rogalev, Nikolay; Osipov, Sergey; Zlyvko, Olga; Naumov, Vladimir (2022)."Research and Development of Trinary Power Cycles".Inventions.7 (3): 56.doi:10.3390/inventions7030056.ISSN 2411-5134.
^Ashley MacLennan (2019-01-09)."A complete beginner's guide to baseball stats: Pitching statistics, and what they mean".Bless You Boys. Retrieved2020-07-30.
^"Stats - Team - Pitching". MLB (Major League Baseball). Retrieved2020-07-30.
^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.C-24 (2):212–215.doi:10.1109/T-C.1975.224188.S2CID 38704739.
^Parhami, Behrooz; McKeown, Michael (2013-11-03). "Arithmetic with binary-encoded balanced ternary numbers".2013 Asilomar Conference on Signals, Systems and Computers. Pacific Grove, California, US. pp. 1130–1133.doi:10.1109/ACSSC.2013.6810470.ISBN 978-1-4799-2390-8.S2CID 9603084.{{cite book}}: CS1 maint: location missing publisher (link)
^Jones, Douglas W. (June 2016)."Binary Coded Ternary and its Inverse".
^Jones, Douglas W. (2015-12-29)."Ternary Data Types for C Programmers".
^Impagliazzo, John; Proydakov, Eduard (2006).Perspectives on Soviet and Russian Computing. First IFIP WG 9.7 Conference, SoRuCom 2006. Petrozavodsk, Russia:Springer.ISBN 978-3-64222816-2.
^Brousentsov, N. P.; Maslov, S. P.; Ramil Alvarez, J.; Zhogolev, E. A."Development of ternary computers at Moscow State University". Retrieved2010-01-20.

External links

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Ternary Arithmetic Archived 2011-05-14 at theWayback Machine
The ternary calculating machine of Thomas Fowler
Ternary Base Conversion – includes fractional part, from Maths Is Fun
Gideon Frieder's replacement ternary numeral system
Visualization of ternary numeral system

v t e Data types
Uninterpreted	Bit Byte Trit Tryte Word Bit array
Numeric	Arbitrary-precision or bignum Complex Decimal Fixed point Block floating point Floating point Reduced precision Minifloat Half precision bfloat16 Single precision Double precision Quadruple precision Octuple precision Extended precision Long double Integer signedness Interval Rational
Pointer	Address physical virtual Reference
Text	Character String null-terminated
Composite	Algebraic data type generalized Array Associative array Class Dependent Equality Inductive Intersection List Object metaobject Option type Product Record or Struct Refinement Set Union tagged
Other	Any type Boolean Bottom type Collection Enumerated type Exception Function type Opaque data type Recursive data type Semaphore Stream Strongly typed identifier Type class Empty type Unit type Void
Related topics	Value Abstract data type Boxing Data structure Generic Kind metaclass Parametric polymorphism Primitive data type Interface Subtyping Type constructor Type conversion Type system Type theory Variable