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Ternary is the base3 numeral system. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit contains <math>\log_2 3</math> (about 1.58496) bits of information. Although ternary most often refers to a system in which the three digits 0, 1, and 2 are all nonnegative numbers, the adjective also lends its name to the balanced ternary system, used in comparison logic and ternary computers.
Contents
Comparison to other radices
*  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 corresponds to binary 101101101 (9 digits) and to ternary 111112 (6 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 
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 
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 
Decimal  19  20  21  22  23  24  25  26  27 
Ternary  1  10  100  1 000  10 000 

Binary  1  11  1001  1 1011  101 0001 
Decimal  1  3  9  27  81 
Power  3^{0}  3^{1}  3^{2}  3^{3}  3^{4} 
Ternary  100 000  1 000 000  10 000 000  100 000 000  1 000 000 000 
Binary  1111 0011  10 1101 1001  1000 1000 1011  1 1001 1010 0001  100 1100 1110 0011 
Decimal  243  729  2 187  6 561  19 683 
Power  3^{5}  3^{6}  3^{7}  3^{8}  3^{9} 
As for rational numbers, ternary offers a convenient way to represent one third (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 one half (neither for one quarter, one sixth, one eighth, one tenth, etc.), because 2 is not a prime factor of the base.
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 
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 maximum 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), and
N(b,d) = b^{d} − 1. For a 3digit 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
A basethree system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a single hand for counting prayers (as alternative for the Misbaha).
In certain analog logic, the state of the circuit is often expressed ternary. This is most commonly seen in Transistor–transistor logic using 7406 open collector logic. 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" is used to denote fractional parts of an inning in baseball. Since each inning consists of three outs, each out is considered one third of an inning and is denoted as .1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the fractional part of the number is written in ternary form.
Ternary numbers can be used to convey selfsimilar 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.^{[1]}^{[2]} 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: .1020 is equivalent to .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 highest radix economy, followed closely by binary and quaternary. It has been used for some computing systems because of this efficiency. It is also used to represent 3 option trees, such as phone menu systems, which allow a simple path to any branch.
A form of Redundant binary representation called Balanced ternary or Signeddigit representation is sometimes used in lowlevel software and hardware to accomplish fast addition of integers because it can eliminate carries.^{[3]}
Tryte
Some ternary computers such as the Setun defined a tryte to be 6 trits, analogous to the binary byte.^{[4]}
See also
Notes
 ^ Mohsen Soltanifar, On A sequence of cantor Fractals, Rose Hulman Undergraduate Mathematics Journal, Vol 7, No 1, paper 9, 2006.
 ^ Mohsen Soltanifar, A Different Description of A Family of Middlea Cantor Sets, American Journal of Undergraduate Research, Vol 5, No 2, pp 9–12, 2006.
 ^ Dhananjay Phatak, I. Koren, Hybrid SignedDigit Number Systems: A Unified Framework for Redundant Number Representations with Bounded Carry Propagation Chains, 1994, [1]
 ^ Brousentsov, N. P.; Maslov, S. P.; Ramil Alvarez, J.; Zhogolev, E.A. "Development of ternary computers at Moscow State University". Retrieved 20 January 2010.
References
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (September 2010) 
 Hayes, Brian (2001), "Third base", American Scientist 89 (6): 490–494, doi:10.1511/2001.40.3268.
External links
 Third Base
 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
 Visualization of numeral systems
