A ternary numeral system (also called base 3 or trinary) has three as its base. 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, balanced ternary is widely used in comparison logic and ternary computers.
Information units
Trit A ternary digit is a trit (trinary digit), analogously to a bit. One trit is equivalent to log2 3 (about 1.58496) bits of information. Under classical hardware assumptions, base-3 is theoretically more efficient than binary in terms of radix economy, as 3 is the closest integer to Euler's number (e).
Tribble Analogous to the binary nibble, a tribble consists of 3 trits. It can hold 27 distinct states (33 = 27), which is roughly equivalent to 4.75 bits of information. Because it has exactly 27 states, a tribble is conveniently represented by a single character in a septemvigesimal (base-27) alphanumeric notation.
Tryte Analogous to a binary byte, a tryte is typically defined as 6 or 9 trits. Early ternary computers, such as the Soviet Setun, defined a tryte to be six trits. A six-trit tryte holds 36 = 729 states, carrying approximately 9.5 bits of information—significantly more than the standard 8-bit binary byte (256 states). Modern theoretical architectures often favor a 9-trit tryte (39 = 19,683 states), as it cleanly divides into three tribbles.
Word A ternary word represents the standard register width for a given ternary architecture. The Setun computer, for instance, operated using an 18-trit word with 9-trit command and memory architectures.
Macro units When scaling to larger data storage capacities, ternary computing nomenclature splits into two distinct conventions depending on the hardware architecture:
Native ternary scaling: In a pure ternary system, macro-units scale by powers of 3 rather than the powers of 2 (1,024) or 10 (1,000) used in binary and decimal systems. A kilotryte (KT) in this system is exactly 310 (59,049) trytes. Following this pattern, a megatryte (MT) is 320 trytes, and a gigatryte (GT) is 330 trytes. Binary-compatible scaling: In hybrid systems designed to interface directly with existing binary infrastructure, engineers often map ternary groupings to standard binary sizes. Because 5 trits (243 states) fit efficiently within an 8-bit byte (256 states), traditional binary SI prefixes are sometimes applied directly to tryte blocks, making a kilotryte equal to 1,024 trytes, and a megatryte equal to 1,048,576 trytes.
Comparison to other bases Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal 36510 or senary 14056 corresponds to binary 1011011012 (nine bits) and to ternary 1111123 (six trits). 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).
As for rational numbers, ternary offers a convenient way to represent 1/3 similar to 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 has a prime factor that is not a factor of the base; as with base two, one-tenth (decimal 1/10, senary 1/14) is not representable exactly (that would need e.g. decimal); nor is one-sixth (senary 1/10, decimal 1/6).
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 ) {\displaystyle N(b,d)} with base b {\displaystyle b} and d {\displaystyle d} digits, all of which are the maximal digit value b − 1 {\displaystyle b-1} , we can write the geometric progression:
N ( b , d ) = ( b − 1 ) b d − 1 + ( b − 1 ) b d − 2 + ⋯ + ( b − 1 ) b 0 = ( b − 1 ) ( b d − 1 + b d − 2 + ⋯ + 1 ) = ( b − 1 ) M . {\displaystyle {\begin{aligned}N(b,d)&=(b-1)b^{d-1}+(b-1)b^{d-2}+\dots +(b-1)b^{0}\\&=(b-1)(b^{d-1}+b^{d-2}+\dots +1)\\&=(b-1)M.\end{aligned}}}
Multiplying M {\displaystyle M} by b {\displaystyle b} yields:
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