Ternary numeral system

A ternary /ˈtɜːrnəri/ numeral system (also called base 3 or trinary) has three as its base. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit is equivalent to log₂ 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

A ternary multiplication 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	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 (base 9) and septemvigesimal (base 27).

Numbers from 1 to 3³ in standard ternary

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

Powers of 3 in ternary

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, one-tenth (decimal1/10, senary 1/14) is not representable exactly (that would need e.g. decimal); nor is one-sixth (senary 1/10, decimal 1/6).

Fractions in ternary

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 2ⁿ − 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)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

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

 

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 in CMOS circuits, and also in transistor–transistor logic with totem-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 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 an 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 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 lowest radix economy, followed closely by binary and quaternary. This is due to its proximity to the mathematical constant e. It has been used for some computing systems because of this efficiency. It is also used to represent three-option trees, such as phone menu systems, which allow a simple path to any branch.

A form of redundant binary representation called a binary signed-digit number system, a form of signed-digit representation, is sometimes used in low-level software and hardware to accomplish fast addition of integers because it can eliminate carries.^[5]

Binary-coded ternary

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 bits used to encode each trit.^[6][7] BCT encoding is analogous to binary-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 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

• Ternary logic
• Tai Xuan Jing
• Setun, a ternary computer
• Qutrit
• Ternary floating point

References

1. Ashley MacLennan (2019-01-09). "A complete beginner's guide to baseball stats: Pitching statistics, and what they mean". Bless You Boys. Retrieved 2020-07-30.
2. "Stats - Team - Pitching". MLB (Major League Baseball). Retrieved 2020-07-30.
3. Soltanifar, Mohsen (2006). "On A sequence of cantor Fractals". Rose Hulman Undergraduate Mathematics Journal. 7 (1). Paper 9.
4. Soltanifar, Mohsen (2006). "A Different Description of A Family of Middle–α Cantor Sets". American Journal of Undergraduate Research. 5 (2): 9–12.
5. 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.
6. 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.
7. Parhami, Behrooz; McKeown, Michael (2013-11-03). "Arithmetic with Binary-Encoded 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 978-1-4799-2390-8. S2CID 9603084.
8. Jones, Douglas W. (June 2016). "Binary Coded Ternary and its Inverse".
9. Jones, Douglas W. (2015-12-29). "Ternary Data Types for C Programmers".
10. Impagliazzo, John; Proydakov, Eduard (2011-09-06). 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 978-3-64222816-2.
11. Brousentsov, N. P.; Maslov, S. P.; Ramil Alvarez, J.; Zhogolev, E. A. "Development of ternary computers at Moscow State University". Retrieved 2010-01-20.

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. (PDF) from the original on 2019-10-30. Retrieved 2020-04-12.

External links

• Ternary Arithmetic 2011-05-14 at the Wayback 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