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Encyclopedia > Ternary numeral system
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Ternary or trinary is the base-3 numeral system. Ternary digits are known as trits (trinary digit), with a name analogous to "bit". Although ternary most often refers to a system in which the three digits, 0, 1, and 2, are all nonnegative integers, the adjective also lends its name to the balanced ternary system, used in comparison logic and ternary computers. 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 counting rods (Traditional Chinese: , Simplified Chinese: , pinyin: chou2) were used by ancient Chinese before the invention of the abacus. ... 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 Āryabhaa. ... 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 Herodian. ... 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. ... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ... 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, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 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, hexadecimal, base-16, 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. ... It has been suggested that Radix-64 be merged into this article or section. ... 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. ... In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ... This article discusses the number three. ... A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ... In mathematics and computer science, a numerical digit is a symbol, e. ... This article is about the unit of information. ... For other uses, see zero or 0. ... Look up one in Wiktionary, the free dictionary. ... “II” redirects here. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Balanced ternary is a non-standard positional numeral system, useful for comparison logic. ... Ternary computers use three-valued logic in their calculations. ...

Contents

Comparison to other radixes

Compared to decimal and binary

Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal 365 corresponds to binary 101101101 (9 digits) and to ternary 111112 (6 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 and septemvigesimal. The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... Previous number: 364 Next number: 366 365 is a semiprime centered square number. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ...

Numbers one to twenty-seven in standard ternary
Ternary 1 2 10 11 12 20 21 22 100
Binary 1 10 11 100 101 110 111 1000 1001
Decimal 1 2 3 4 5 6 7 8 9
Ternary 101 102 110 111 112 120 121 122 200
Binary 1010 1011 1100 1101 1110 1111 10000 10001 10010
Decimal 10 11 12 13 14 15 16 17 18
Ternary 201 202 210 211 212 220 221 222 1000
Binary 10011 10100 10101 10110 10111 11000 11001 11010 11011
Decimal 19 20 21 22 23 24 25 26 27
Powers of three in ternary
Ternary 1 10 100 1 000 10 000
Binary 1 11 1001 1 1011 101 0001
Decimal 1 3 9 27 81
Power 30 31 32 33 34
Ternary 100 000 1 000 000 10 000 000 100 000 000 1 000 000 000
Binary 1111 0011 10 1101 1001 1000 1000 1011 1 1001 1010 0001 100 1100 1110 0011
Decimal 243 729 2 187 6 561 19 683
Power 35 36 37 38 39

As for rational numbers, ternary offers a convenient way to represent one third (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 the most basic fraction: one half (and thus, neither for one quarter, one sixth, one eighth, one tenth, etc.), because 2 is not a prime factor of the base. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... “II” redirects here. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... ...

Fractions in ternary
Ternary 0.111111111111... 0.1 0.020202020202... 0.012101210121... 0.011111111111... 0.010212010212...
Binary 0.1 0.010101010101... 0.01 0.001100110011... 0.00101010101... 0.001001001001...
Decimal 0.5 0.333333333333... 0.25 0.2 0.166666666666... 0.142857142857...
Fraction 1/2 1/3 1/4 1/5 1/6 1/7
Ternary 0.010101010101... 0.01 0.002200220022... 0.002110021100... 0.002020202020... 0.002002002002...
Binary 0.001 0.000111000111... 0.000110011001... 0.000101110100... 0.000101010101... 0.000100111011...
Decimal 0.125 0.111111111111... 0.1 0.090909090909... 0.083333333333... 0.076923076923...
Fraction 1/8 1/9 1/10 1/11 1/12 1/13

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) is often used, similar to how octal and hexadecimal systems are used in place of binary. Ternary also has a unit similar to a byte, the tryte, which is six ternary digits. Nonary is a base 9 numeral system, typically using the digits 0-8, but not the digit 9. ... A Septemvigesimal numeral system has a base of twenty-seven. ... 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, hexadecimal, base-16, 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. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... In computer science a byte is a unit of measurement of information storage, most often consisting of eight bits. ...


Practical usage

A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a single hand for counting prayers (as alternative for the Misbaha). The benefit —apart from allowing a single hand to count up to 99 or to 100— is that counting doesn't distract the mind too much since the counter needs only to divide Tasbihs into groups of three. For people named Islam, see Islam (name). ... Tasbih (تسبيح) is a form of dhikr that involves the repetitive utterances of short sentences glorifying god. ... The hands (med. ... A Misbaha A Misbaha (Arabic: مسبحة) or Subha (Arabic:سبحة) is a string of prayer beads, probably of Persian origin, which is traditionally used by Muslims to keep track of counting in Tasbih. ... Tasbih (تسبيح) is a form of dhikr that involves the repetitive utterances of short sentences glorifying god. ...


A rare ternary point is used to denote fractional parts of an inning in baseball. Since each inning consists of 3 outs, each out is considered begin{matrix} frac{1}{3} end{matrix} (one third) of an inning and is denoted as .1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3 begin{matrix} frac{2}{3} end{matrix}. (In this usage, only the fractional part of the number is written in ternary form.) An innings, or inning, is a fixed-length segment of a game in any of a variety of sports – most notably baseball and cricket – during which one team attempts to score while the other team attempts to prevent the first from scoring. ... This article is about the sport. ... In baseball, an out occurs when the defensive team effects any of a number of different events, and the umpire rules a batter or baserunner out. ... 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 in the game. ...


Ternary numbers can be used to convey self-similar structures like a Sierpinski Triangle or a Cantor set conveniently. Sierpinski triangle The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Wacław Sierpiński who described it in 1916. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...


Ternary encoding can also be applied in most Relational Databases, with NULL acting as the third trit. By manually encoding datatypes it is possible to effectively reduce the storage requirements by a third.


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