Balanced ternary is a non-standard positional numeral system, useful for comparison logic. It is a ternary system, but unlike the standard (unbalanced) ternary system, the digits have the values −1, 0, and 1. This combination is especially valuable for ordinal relationships between two values, where the three possible relationships are less-than, equals, and greater-than. Balanced ternary can represent all integers without resorting to a separate minus sign. A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ... The Hindu-Arabic numeral system (also called Algorism) is a positional decimal numeral system documented from the 9th century. ... Numerals sans-serif Arabic numerals, known formally as Hindu-Arabic numerals, and also as Indian numerals, Hindu numerals, Western Arabic numerals, European numerals, or Western numerals, are the most common symbolic representation of numbers around the world. ... The Eastern Arabic numerals (also called Eastern Arabic numerals, Arabic-Indic numerals, Arabic Eastern Numerals) are the symbols (glyphs) used to represent the Hindu-Arabic numeral system in conjunction with the Arabic alphabet in Egypt, Iran, Pakistan and parts of India, and also in the no longer used Ottoman Turkish... Khmer numerals are the numerals used in the Khmer language of Cambodia. ... India has produced many numeral systems. ... The Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). ... The Abjad numerals are a decimal numeral system which was used in the Arabic-speaking world prior to the use of the Hindu-Arabic numerals from the 8th century, and in parallel with the latter until Modern times. ... Cyrillic numerals was a numbering system derived from the Cyrillic alphabet, used by South and East Slavic peoples. ... Note: This article contains special characters. ... The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. ... Greek numerals are a system of representing numbers using letters of the Greek alphabet. ... The Sanskrit alphabetic numerals were created in about A.D. 510 by Ä€ryabhaa. ... Attic numerals were used by ancient Greeks, possibly from the 7th century BC. They were also known as Herodianic numerals because they were first described in a 2nd century manuscript by Herodianus. ... The Etruscan numerals were used by the ancient Etruscans. ... During the beginning of the Urnfield culture, around 1200 BC, a series of votive sickles of bronze with marks that have been interpreted as a numeral system, appeared in Central Europe. ... Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ... Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. ... Mayan numerals. ... This is a list of numeral system topics, by Wikipedia page. ... Positional notation is a system in which each position has a value represented by a unique symbol or character. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ... Quaternary is the base four numeral system. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... In mathematics and computer science, base-16, hexadecimal, or simply hex, is a numeral system with a radix or base of 16, usually written using the symbols 0–9 and A–F or a–f. ... Base32 is a derivation of Base64 with the following additional properties: The resulting character set is all uppercase, which can often be beneficial when using a case-sensitive filesystem. ... Base 64 is a positional numeral system using a base of 64. ... Ternary or trinary is the base-3 numeral system. ... Nonary is a base 9 numeral system, typically using the digits 0-8, but not the digit 9. ... The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. ... As there are 24 hours in a day a numbering system based upon 24, and as the base 12 is convenient here some examples of the base 24 (quadrovigesimal) system. ... Base 30 or trigesimal is a positional numeral system using 30 as the radix. ... Base 36 refers to a positional numeral system using 36 as the radix. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... Non-standard positional numeral systems are numeral systems that may be denoted positional systems, but that deviate in one way or another from the following description of standard positional system: In a standard positional numeral system, the base b is a positive integer, and b different glyphs are used to... Ternary or trinary is the base-3 numeral system. ... The plus (+) and minus (−) signs are used universally to represent the operations of addition and subtraction, and have been extended to many other meanings, more or less analogous. ...

Balanced ternary is counted as follows. (In this example, the symbol 1 denotes the digit −1, but alternatively for easier parsing "−" may be used to denote −1 and "+" to denote +1.)

Unbalanced ternary can be converted to balanced ternary notation by adding 1111.. with carry, then subtracting 1111... without borrow. For example, 021₃ + 111₃ = 202₃, 202₃ − 111₃ = 111_3(bal) = 7₁₀.

Balanced ternary is easily represented as electronic signals, as potential can either be negative, neutral, or positive. Utilizing a third state encompasses more data per digit; linearly approximately log(3)/log(2)=~1.5849 bits per trit. Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

Applications

Balanced ternary has other applications besides computing. For example, a classical "2-pan" balance, with one weight for each power of 3, can weigh relatively heavy objects accurately with a small number of weights, by moving weights between the two pans and the table. For example, with weights for each power of 3 through 81, a 60-gram object (60₁₀ = 11110) will be balanced perfectly with an 81 gram weight in the other pan, the 27 gram weight in its own pan, the 9 gram weight in the other pan, the 3 gram weight in its own pan, and the 1 gram weight set aside. This is an optimal solution in terms of the number of weights needed to weigh any object. For other meanings of the word balance, see: propaganda equilibrium (disambiguation page) sense of balance weighing scale analytical balance (a precise weighing scale) balance beam in gymnastics Balance (song) homeostasis, the biological balance within a human or other animals body When the weights on the plates of this balance...

Similarly, a currency system using ternary values would save visits to the bank - customers would be likely to have exact change, or be able to get a small number of coins in change, and sellers would just occasionally need to deposit a large coin or two. The system works by representing positive values as coins the customer gives the merchant, and negative values as coins the merchant gives the customer. For example, if a merchant sells an item for five zorkmids, the customer would give the merchant a nine-zorkmid coin, and the merchant would give the customer a three-zorkmid coin and a one-zorkmid coin. Zork universe Zork games Zork Anthology Zork trilogy Zork I   Zork II   Zork III Beyond Zork   Zork Zero   Planetfall Enchanter trilogy Enchanter   Sorcerer   Spellbreaker Other games Wishbringer   Return to Zork Zork: Nemesis   Zork Grand Inquisitor Zork: The Undiscovered Underground Topics in Zork Encyclopedia Frobozzica Characters   Kings   Creatures Timeline   Magic   Calendar...

Computation

There were a few experimental Russian computers in the early days of computing that were built with balanced ternary instead of binary. The notation has a number of computational advantages over regular ternary. Particularly, the one-digit multiplication table has no carries in balanced ternary, and the addition table has only two symmetric carries instead of three. In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. ...

This notation has the property that the leading non-zero digit is the sign of the full number. To compare two numbers, simply compare digits from the most significant to the least significant. The direction of the magnitude compare of the first digits that are different is the direction of the magnitude compare of the full numbers.

A number is divisible by three if the last digit is zero. The quick test for even is the analog of the base ten divide-by-nine test: add up all the digits and repeat until you have a one-digit number; the number is even if the final sum is zero.

Fractional balanced ternary

Balanced ternary can be extended to fractional numbers similarly to how decimal numbers are written past the decimal point.[1] For instance, ⅓ is 0.1 and ⅔ is 1.1. This article or section does not cite its references or sources. ...

Donald Knuth has pointed out that truncation and rounding are the same operation in balanced ternary—they produce exactly the same result. Moreover, there is no ambiguity in rounding (a property shared with other odd bases) since the number ½ is represented by the repeating fraction 0.1… (and 1.1…). Donald Knuth at a reception for the Open Content Alliance. ...

Unlike in standard base notation, where integer values and terminating fractions have multiple representations (e.g. in decimal, ¼ = 0.25₁₀ = 0.249…₁₀), in balanced ternary such numbers have only one representation (¼ = 0.(11)…_3(bal)). On the other hand, numbers half of a terminating fraction (i.e. whose denominator is 2 times a power of 3) do have multiple representations e.g. ⅙ = 0.0(1)…_3(bal) = 0.1(1)…_3(bal).

The basic operations—addition, subtraction, multiplication, and division—are done as in regular ternary, except that there are some special considerations on the size compare. Multiply 2 by 10 in ternary and then divide the result by 2 to see the issue. The intermediate results are the same (in reverse order) in the two cases, and so you can use one as a check on the other.

Digit shifts multiply or divide by three instead of by two as in binary.

Multiplication by two can be done by adding a number to itself. Division by two can be done with the same operation count as an add by LeRoy Eide's algorithm (an algorithm that returns the result least significant digit first, instead of most significant digit first as standard division).

A generalization of LeRoy Eide's algorithm provides fast (same overhead as an add) algorithms for division by any number of the form (). (Which includes fast divide algorithms for the popular numbers 2, 4, 8, and 10 [among others])

As in binary, there are balanced ternary equivalents of shift and add multiplication, and shift and multiply exponentiation algorithms.

A common convention for balanced ternary is to represent the digits by "+" for +1, "−" for −1, and "0" for zero. Using this convention, the multiplication and addition tables are:

See also

• Setun, a ternary computer
• Ternary logic
• Ternary computer

Setun (Russian: ) was a balanced ternary computer developed in 1958 in Russia. ... A ternary, three-valued or trivalent logic is a term to describe any of several multi-valued logic systems in which there are three truth values indicating true, false and some third value. ... Ternary computers use three-valued logic in their calculations. ...

External links

• Development of ternary computers at Moscow State University
• Nikolay Brusentsov
• Representation of Fractional Numbers in Balanced Ternary