A ternary
numeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner.
The same s ...
(also called base 3 or trinary) has
three
3 is a number, numeral, and glyph.
3, three, or III may also refer to:
* AD 3, the third year of the AD era
* 3 BC, the third year before the AD era
* March, the third month
Books
* '' Three of Them'' (Russian: ', literally, "three"), a 1901 ...
as its
base. Analogous to a
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
, a ternary
digit is a trit (trinary digit). One trit is equivalent to
log2 3 (about 1.58496) bits of
information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...
.
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
Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1. This stands in contrast to the standard (unbalance ...
system; comprising the digits
−1
In mathematics, −1 (also known as negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less t ...
, 0 and +1, used in comparison logic and
ternary computer
A ternary computer, also called trinary computer, is one that uses ternary logic (i.e., base 3) instead of the more common binary system (i.e., base 2) in its calculations. This means it uses trits (instead of bits, as most computers do).
Types ...
s.
Comparison to other bases
Representations of
integer number
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s in ternary do not get uncomfortably lengthy as quickly as in
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
. For example,
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
365 or
senary
A senary () numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like decimal, it is a semiprime, though it is unique as the product of the only two con ...
1405 corresponds to binary 101101101 (nine digits) and to ternary 111112 (six digits). However, they are still far less compact than the corresponding representations in bases such as
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
see below for a compact way to codify ternary using nonary (base 9) and
septemvigesimal
There are many different numeral systems, that is, writing systems for expressing numbers.
By culture / time period
By type of notation
Numeral systems are classified here as to whether they use positional notation (also known as place-value ...
(base 27).
As for
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, ternary offers a convenient way to represent 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 (nor for , , etc.), because
2 is not a
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
factor
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, suc ...
of the base; as with base two, one-tenth (decimal, senary ) is not representable exactly (that would need e.g. decimal); nor is one-sixth (senary , decimal ).
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''(''b'', ''d'') with base ''b'' and ''d'' digits, all of which are the maximal digit value , we can write:
:
:
:.
: and
:, so
:, or
:
Then
:
:
:
For a three-digit ternary number, .
Compact ternary representation: base 9 and 27
Nonary (base 9, each digit is two ternary digits) or
septemvigesimal
There are many different numeral systems, that is, writing systems for expressing numbers.
By culture / time period
By type of notation
Numeral systems are classified here as to whether they use positional notation (also known as place-value ...
(base 27, each digit is three ternary digits) can be used for compact representation of ternary, similar to how
octal
The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...
and
hexadecimal
In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexa ...
systems are used in place of
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
.
Practical usage
![fewest_weights_balance_puzzle](https://upload.wikimedia.org/wikipedia/commons/7/78/Fewest_weights_balance_puzzle.svg)
In certain analog logic, the state of the circuit is often expressed ternary. This is most commonly seen in
CMOS
Complementary metal–oxide–semiconductor (CMOS, pronounced "sea-moss", ) is a type of metal–oxide–semiconductor field-effect transistor (MOSFET) fabrication process that uses complementary and symmetrical pairs of p-type and n-type MOSFE ...
circuits, and also in
transistor–transistor logic Transistor–transistor logic (TTL) is a logic family built from bipolar junction transistors. Its name signifies that transistors perform both the logic function (the first "transistor") and the amplifying function (the second "transistor"), as opp ...
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
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
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
Baseball is a bat-and-ball sport played between two teams of nine players each, taking turns batting and fielding. The game occurs over the course of several plays, with each play generally beginning when a player on the fielding tea ...
(usually just for
pitchers
In baseball, the pitcher is the player who throws ("pitches") the baseball from the pitcher's mound toward the catcher to begin each play, with the goal of retiring a batter, who attempts to either make contact with the pitched ball or draw ...
), to denote fractional parts of an inning. Since the team on offense is allowed three
outs
In baseball, an out occurs when the umpire rules a batter or baserunner out. When a batter or runner is out, they lose their ability to score a run and must return to the dugout until their next turn at bat. When three outs are recorded in a ha ...
, 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
In baseball, innings pitched (IP) are the number of innings a pitcher has completed, measured by the number of batters and baserunners that are put out while the pitcher is on the pitching mound in a game. Three outs made is equal to one innin ...
column for that game would be listed as 3.2, the equivalent of (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
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Thr ...
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
The radix economy of a number in a particular base (or radix) is the number of digits needed to express it in that base, multiplied by the base (the number of possible values each digit could have). This is one of various proposals that have been ...
, followed closely by
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
and
quaternary
The Quaternary ( ) is the current and most recent of the three periods of the Cenozoic Era in the geologic time scale of the International Commission on Stratigraphy (ICS). It follows the Neogene Period and spans from 2.58 million years ...
. This is due to its proximity to the
mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
''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 A redundant binary representation (RBR) is a numeral system that uses more bits than needed to represent a single binary Numerical digit, digit so that most numbers have several representations. An RBR is unlike usual binary numeral systems, includi ...
called a binary signed-digit number system, a form of
signed-digit representation
In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers.
Signed-digit representation can be used to accomplish fast addition of integers because ...
, 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 bits used to encode each trit.
BCT encoding is analogous to
binary-coded decimal
In computing and electronic systems, binary-coded decimal (BCD) is a class of binary encodings of decimal numbers where each digit is represented by a fixed number of bits, usually four or eight. Sometimes, special bit patterns are used for ...
(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
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...
.
A library of
C code supporting BCT arithmetic is available.
Tryte
Some
ternary computer
A ternary computer, also called trinary computer, is one that uses ternary logic (i.e., base 3) instead of the more common binary system (i.e., base 2) in its calculations. This means it uses trits (instead of bits, as most computers do).
Types ...
s such as the
Setun
Setun (russian: Сетунь) was a computer developed in 1958 at Moscow State University. It was built under the leadership of Sergei Sobolev and Nikolay Brusentsov. It was the most modern ternary computer, using the balanced ternary numeral sys ...
defined a tryte to be six trits
or approximately 9.5
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s (holding more information than the ''de facto''
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
).
See also
*
Ternary logic
In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating ''true'', ''false'' and some indeterminate ...
*
Tai Xuan Jing
The text ''Tài Xuán Jīng'' ("Canon of Supreme Mystery", ) is a guide for divination composed by the Confucian writer Yang Xiong (53 BCE – 18 CE). The first draft of this work was completed in 2 BCE (in the decade before the fall of the Wes ...
*
Setun
Setun (russian: Сетунь) was a computer developed in 1958 at Moscow State University. It was built under the leadership of Sergei Sobolev and Nikolay Brusentsov. It was the most modern ternary computer, using the balanced ternary numeral sys ...
, a
ternary computer
A ternary computer, also called trinary computer, is one that uses ternary logic (i.e., base 3) instead of the more common binary system (i.e., base 2) in its calculations. This means it uses trits (instead of bits, as most computers do).
Types ...
*
Qutrit
A qutrit (or quantum trit) is a unit of quantum information that is realized by a 3-level quantum system, that may be in a superposition of three mutually orthogonal quantum states.
The qutrit is analogous to the classical radix-3 trit, just as ...
*
Ternary floating point
References
Further reading
*
External links
Ternary Arithmetic
Ternary Base Conversionncludes fractional part, from Maths Is Fun
Gideon Frieder's replacement ternary numeral system
{{DEFAULTSORT:Ternary Numeral System
Computer arithmetic
Positional numeral systems
Ternary computers