The ternarynumeral system has three as its base. Analogous to a bit, a ternary digit is a trit. One trit is equivalent to log2 3 bits of information. Although ternary most often refers to a system in which the three digits are all non–negative numbers, specifically,, and, 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
Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal365 or senary 1405 corresponds to binary 101101101 and to ternary 111112. However, they are still far less compact than the corresponding representations in bases such as decimalsee 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
30
31
32
33
34
-
-
-
-
-
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 for rational numbers, ternary offers a convenient way to represent as same as senary ; but a major drawback is that, in turn, ternary does not offer a finite representation for , because 2 is not a prime factor of the base; as with base two, one-tenth is not representable exactly ; nor is one-sixth.
Fraction
'
'
'
'
'
'
'
'
'
'
'
'
Ternary
0.
0.1
0.
0.
0.0
0.
0.
0.01
0.
0.
0.0
0.
Binary
0.1
0.
0.01
0.
0.0
0.
0.001
0.
0.0
0.
0.00
0.
Senary
0.3
0.2
0.13
0.
0.1
0.
0.043
0.04
0.0
0.
0.03
0.
Decimal
0.5
0.
0.25
0.2
0.1
0.
0.125
0.
0.1
0.
0.08
0.
Sum of the digits in ternary as opposed to binary
The value of a binary number with n bits that are all 1 is. Similarly, for a number N with base b and d digits, all of which are the maximal digit value, we can write: Then For a three-digit ternary number,.
Compact ternary representation: base 9 and 27
Nonary or septemvigesimal 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 totem-pole output. The output is said to either be low, high, or open. 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 statics in American baseball, 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 . In this usage, only the fractional part of the number is written in ternary form. 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. 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 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 andquaternary. 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.
Binary-coded ternary
Simulation of ternary computers using binary computers, or interfacing between ternary and binary computers, can involve use of binary-coded ternary numbers, with two bits used to encode each trit. BCT encoding is analogous to binary-coded decimal 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. A library of C code supporting BCT arithmetic is available.
Tryte
Some ternary computers such as the Setun defined a tryte to be six trits or approximately 9.5 bits.
