Ternary numeral system
A ternary numeral 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, decimal 365 or senary corresponds to binary and to ternary . 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.| × | 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 |
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.
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 statistics 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 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.