Ternary numeral system
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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 log_{2} 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
× | 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_{(10)} or senary 1405_{(6)} corresponds to binary 101101101_{(2)} (nine bits) and to ternary 111112_{(3)} (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 3^{3} − 1 in standard ternary Ternary 0 1 2 10 11 12 20 21 22 Binary 0 1 10 11 100 101 110 111 1000 Senary 0 1 2 3 4 5 10 11 12 Decimal 0 1 2 3 4 5 6 7 8 Ternary 100 101 102 110 111 112 120 121 122 Binary 1001 1010 1011 1100 1101 1110 1111 10000 10001 Senary 13 14 15 20 21 22 23 24 25 Decimal 9 10 11 12 13 14 15 16 17 Ternary 200 201 202 210 211 212 220 221 222 Binary 10010 10011 10100 10101 10110 10111 11000 11001 11010 Senary 30 31 32 33 34 35 40 41 42 Decimal 18 19 20 21 22 23 24 25 26
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 2^{n} − 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)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,
- N(b, d) = b^{d} − 1.
For a three-digit 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
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.
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.
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. 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. 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 (holding more information than the de facto binary byte).