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.

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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
×12101112202122100
1 12101112202122100
2 2112022101110112121200
10 10201001101202002102201000
11 1122110121202220100110121100
12 121011202022211010102211111200
20 2011020022010101100112012102000
21 21112210100110221120121120022100
22 22121220101211111210200221012200
100 10020010001100120020002100220010000

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 0 to 33 − 1 in standard ternary
Ternary 012101112202122
Binary 0110111001011101111000
Senary 012345101112
Decimal 012345678
Ternary 100101102110111112120121122
Binary 10011010101111001101111011111000010001
Senary 131415202122232425
Decimal 91011121314151617
Ternary 200201202210211212220221222
Binary 100101001110100101011011010111110001100111010
Senary 303132333435404142
Decimal 181920212223242526
Powers of 3 in ternary
Ternary 110100100010000
Binary 1111001110111010001
Senary 131343213
Decimal 1392781
Power 3031323334
Ternary 1000001000000100000001000000001000000000
Binary 1111001110110110011000100010111100110100001100110011100011
Senary 104332131404350213231043
Decimal 2437292187656119683
Power 3536373839

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 1/21/31/41/51/61/71/81/91/101/111/121/13
Ternary 0.10.10.020.01210.010.0102120.010.010.00220.002110.0020.002
Binary 0.10.010.010.00110.0010.0010.0010.0001110.000110.00010111010.00010.000100111011
Senary 0.30.20.130.10.10.050.0430.040.030.03134524210.030.024340531215
Decimal 0.50.30.250.20.160.1428570.1250.10.10.090.0830.076923

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)bd−1 + (b − 1)bd−2 + … + (b − 1)b1 + (b − 1)b0,
N(b, d) = (b − 1)(bd−1 + bd−2 + … + b1 + 1),
N(b, d) = (b − 1)M.
bM = bd + bd−1 + … + b2 + b1 and
−M = −bd−1 − bd−2 − ... − b1 − 1, so
bM − M = bd − 1, or
M = bd − 1/b − 1.

Then

N(b, d) = (b − 1)M,
N(b, d) = (b − 1)(bd − 1)/b − 1,
N(b, d) = bd − 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 or three 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

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". 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)
  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 (2006). Perspectives on Soviet and Russian Computing. First IFIP WG 9.7 Conference, SoRuCom 2006. Petrozavodsk, Russia: 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
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