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The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
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 ...
using twelve as its base. The number twelve (that is, the number written as "12" in the
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 ...
numerical system) is instead written as "10" in duodecimal (meaning "1
dozen A dozen (commonly abbreviated doz or dz) is a grouping of twelve. The dozen may be one of the earliest primitive integer groupings, perhaps because there are approximately a dozen cycles of the Moon, or months, in a cycle of the Sun, or year ...
and 0 units", instead of "1 ten and 0 units"), whereas the digit string "12" means "1 dozen and 2 units" (decimal 14). Similarly, in duodecimal, "100" means "1  gross", "1000" means "1 
great gross 1728 is the natural number following 1727 and preceding 1729. 1728 is a dozen gross, one great gross (or grand gross). It is the number of cubic inches in a cubic foot. It is also the number of daily chants of the Hare Krishna mantra by a Har ...
", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth", respectively). Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses and , as in
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 ...
, which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, , , 10. The Dozenal Societies of America and Great Britain (organisations promoting the use of duodecimal) use turned digits in their published material: ↊ (a turned 2) for ten and ↋ (a turned 3) for eleven. The number twelve, a
superior highly composite number In mathematics, a superior highly composite number is a natural number which has the highest ratio of its number of divisors to ''some'' positive power of itself than any other number. It is a stronger restriction than that of a highly composite ...
, is the smallest number with four non-trivial
factors 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 ...
(2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the
subitizing Subitizing is the rapid, accurate, and confident judgments of numbers performed for small numbers of items. The term was coined in 1949 by E. L. Kaufman et al., and is derived from the Latin adjective '' subitus'' (meaning "sudden") and captures ...
range, and the smallest
abundant number In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The ...
. All multiples of reciprocals of 3-smooth numbers ( where are integers) have a terminating representation in duodecimal. In particular,  (0.3),  (0.4),  (0.6),  (0.8), and  (0.9) all have a short terminating representation in duodecimal. There is also higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system. In these respects, duodecimal is considered superior to decimal (which has only 2 and 5 as factors) and other proposed bases like
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, ...
or
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 ...
.
Sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
does even better in this respect (the reciprocals of all 5-smooth numbers terminate), but at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.


Origin

:''In this section, numerals are based on decimal
places Place may refer to: Geography * Place (United States Census Bureau), defined as any concentration of population ** Census-designated place, a populated area lacking its own municipal government * "Place", a type of street or road name ** Often ...
. For example, 10 means ten, and 12 means twelve.'' Languages using duodecimal number systems are uncommon. Languages in the
Nigeria Nigeria ( ), , ig, Naìjíríyà, yo, Nàìjíríà, pcm, Naijá , ff, Naajeeriya, kcg, Naijeriya officially the Federal Republic of Nigeria, is a country in West Africa. It is situated between the Sahel to the north and the Gulf o ...
n Middle Belt such as Janji, Gbiri-Niragu (Gure-Kahugu),
Piti Piti may refer to: * Pīti, a mental factor in Buddhism * PITI, the principal, interest, taxes, and insurance sum of a mortgage payment * Piti (food), a soup dish of Central Asia * Piti (footballer) (born 1981), Spanish footballer * Piti, Guam * P ...
, and the Nimbia dialect of
Gwandara Gwandara is a West Chadic language, and the closest relative of Hausa. Its several dialects are spoken in northern Nigeria, predominantly in the north central region of Nigeria by the Gwandara people and some settlers who are about 30,000 people. ...
; and the
Chepang language Chepang may refer to: *Chepang people, a group indigenous to the lands of Nepal *Chepang language Chepang may refer to: * Chepang people, a group indigenous to the lands of Nepal * Chepang language, the language of the Chepang people {{dab ...
of
Nepal Nepal (; ne, नेपाल ), formerly the Federal Democratic Republic of Nepal ( ne, सङ्घीय लोकतान्त्रिक गणतन्त्र नेपाल ), is a landlocked country in South Asia. It is mai ...
are known to use duodecimal numerals.
Germanic languages The Germanic languages are a branch of the Indo-European language family spoken natively by a population of about 515 million people mainly in Europe, North America, Oceania and Southern Africa. The most widely spoken Germanic language, Engli ...
have special words for 11 and 12, such as ''eleven'' and ''twelve'' in
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...
. They come from
Proto-Germanic Proto-Germanic (abbreviated PGmc; also called Common Germanic) is the reconstructed proto-language of the Germanic branch of the Indo-European languages. Proto-Germanic eventually developed from pre-Proto-Germanic into three Germanic branc ...
*''ainlif'' and *''twalif'' (meaning, respectively ''one left'' and ''two left''), suggesting a decimal rather than duodecimal origin. However,
Old Norse Old Norse, Old Nordic, or Old Scandinavian, is a stage of development of North Germanic languages, North Germanic dialects before their final divergence into separate Nordic languages. Old Norse was spoken by inhabitants of Scandinavia and t ...
used a hybrid decimal/duodecimal counting system, with its words for "one hundred and eighty" meaning 200 and "two hundred" meaning 240. On British Isles, this style of counting survived well into the middle ages as the
long hundred The long hundred, also known as the great hundred or twelfty, is the number 120 (in base-10 Arabic numerals) that was referred to as "hundred" in Germanic languages prior to the 15th century, and is now known as one hundred twenty, or six score. ...
. Historically,
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...
of
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
in many
civilization A civilization (or civilisation) is any complex society characterized by the development of a state, social stratification, urbanization, and symbolic systems of communication beyond natural spoken language (namely, a writing system). Ci ...
s are duodecimal. There are twelve signs of the
zodiac The zodiac is a belt-shaped region of the sky that extends approximately 8° north or south (as measured in celestial latitude) of the ecliptic, the Sun path, apparent path of the Sun across the celestial sphere over the course of the year. ...
, twelve months in a year, and the
Babylonians Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. ...
had twelve hours in a day (although at some point this was changed to 24). Traditional
Chinese calendar The traditional Chinese calendar (also known as the Agricultural Calendar 曆; 农历; ''Nónglì''; 'farming calendar' Former Calendar 曆; 旧历; ''Jiùlì'' Traditional Calendar 曆; 老历; ''Lǎolì'', is a lunisolar calendar ...
s, clocks, and compasses are based on the twelve
Earthly Branches The twelve Earthly Branches or Terrestrial Branches are a Chinese ordering system used throughout East Asia in various contexts, including its ancient dating system, astrological traditions, zodiac and ordinals. Origin This system was built ...
or 24 (12×2)
Solar term A solar term is any of twenty-four periods in traditional Chinese lunisolar calendars that matches a particular astronomical event or signifies some natural phenomenon. The points are spaced 15° apart along the ecliptic and are used by lunisolar ...
s. There are 12 inches in an imperial foot, 12 
troy Troy ( el, Τροία and Latin: Troia, Hittite language, Hittite: 𒋫𒊒𒄿𒊭 ''Truwiša'') or Ilion ( el, Ίλιον and Latin: Ilium, Hittite language, Hittite: 𒃾𒇻𒊭 ''Wiluša'') was an ancient city located at Hisarlik in prese ...
ounces in a troy pound, 12  old British pence in a
shilling The shilling is a historical coin, and the name of a unit of modern currencies formerly used in the United Kingdom, Australia, New Zealand, other British Commonwealth countries and Ireland, where they were generally equivalent to 12 pence o ...
, 24 (12×2) hours in a day, and many other items counted by the
dozen A dozen (commonly abbreviated doz or dz) is a grouping of twelve. The dozen may be one of the earliest primitive integer groupings, perhaps because there are approximately a dozen cycles of the Moon, or months, in a cycle of the Sun, or year ...
, gross (
144 144 may refer to: * 144 (number), the natural number following 143 and preceding 145 * AD 144, a year of the Julian calendar, in the second century AD * 144 BC, a year of the pre-Julian Roman calendar * 144 (film), ''144'' (film), a 2015 Indian com ...
,
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
of 12), or
great gross 1728 is the natural number following 1727 and preceding 1729. 1728 is a dozen gross, one great gross (or grand gross). It is the number of cubic inches in a cubic foot. It is also the number of daily chants of the Hare Krishna mantra by a Har ...
( 1728,
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
of 12). The Romans used a fraction system based on 12, including the uncia which became both the English words ''
ounce The ounce () is any of several different units of mass, weight or volume and is derived almost unchanged from the , an Ancient Roman units of measurement, Ancient Roman unit of measurement. The #International avoirdupois ounce, avoirdupois ounce ...
'' and ''inch''. Pre-
decimalisation Decimalisation or decimalization (see spelling differences) is the conversion of a system of currency or of weights and measures to units related by powers of 10. Most countries have decimalised their currencies, converting them from non-decimal ...
,
Ireland Ireland ( ; ga, Éire ; Ulster Scots dialect, Ulster-Scots: ) is an island in the Atlantic Ocean, North Atlantic Ocean, in Northwestern Europe, north-western Europe. It is separated from Great Britain to its east by the North Channel (Grea ...
and the
United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Europe, off the north-western coast of the continental mainland. It comprises England, Scotland, Wales and North ...
used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the
pound sterling Sterling (abbreviation: stg; Other spelling styles, such as STG and Stg, are also seen. ISO code: GBP) is the currency of the United Kingdom and nine of its associated territories. The pound ( sign: £) is the main unit of sterling, and t ...
or
Irish pound The pound (Irish: ) was the currency of the Republic of Ireland until 2002. Its ISO 4217 code was IEP, and the symbol was £ (or IR£ for distinction). The Irish pound was replaced by the euro on 1 January 1999. Euro currency did not begin cir ...
), and
Charlemagne Charlemagne ( , ) or Charles the Great ( la, Carolus Magnus; german: Karl der Große; 2 April 747 – 28 January 814), a member of the Carolingian dynasty, was King of the Franks from 768, King of the Lombards from 774, and the first Holy ...
established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places. The importance of 12 has been attributed to the number of lunar cycles in a year as well as the fact that humans have 12 finger bones (
phalanges The phalanges (singular: ''phalanx'' ) are digital bones in the hands and feet of most vertebrates. In primates, the thumbs and big toes have two phalanges while the other digits have three phalanges. The phalanges are classed as long bones. ...
) on one hand (three in each of four fingers). It is possible to count to 12 with the thumb acting as a pointer, touching each finger bone in turn. A traditional
finger counting Finger-counting, also known as dactylonomy, is the act of counting using one's fingers. There are multiple different systems used across time and between cultures, though many of these have seen a decline in use because of the spread of Arabic n ...
system still in use in many regions of Asia works in this way and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20, and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i.e. the 60, are full. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk.


Notations and pronunciations

In a numbering system, the base (twelve for duodecimal) must be written as 10, but there are numerous proposals for how to write the quantities (counting values) "ten" and "eleven".


Transdecimal symbols

To allow entry on typewriters, letters such as (as in
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 ...
), (initials of ''Ten'' and ''Eleven''), (X from the
Roman numeral Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...
for ten), or are used. Some employ Greek letters such as (from Greek 'ten' and 'eleven'), or . Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his book ''New Numbers'' (script capital E, ). Edna Kramer in her 1951 book ''The Main Stream of Mathematics'' used a (
sextile In astrology, an aspect is an angle that planets make to each other in the Horoscope; as well as to the Ascendant, Midheaven, Descendant, Lower Midheaven, and other points of astrological interest. As viewed from Earth, aspects are measured by ...
or six-pointed asterisk, hash or octothorpe). The symbols were chosen because they were available on some typewriters; they are also on
push-button telephone The push-button telephone is a telephone that has buttons or keys for dialing a telephone number, in contrast to having a rotary dial as in earlier telephone instruments. Western Electric experimented as early as 1941 with methods of using mec ...
s. This notation was used in publications of the Dozenal Society of America (DSA) from 1974 to 2008. From 2008 to 2015, the DSA used , the symbols devised by
William Addison Dwiggins William Addison Dwiggins (June 19, 1880 – December 25, 1956), was an American type designer, calligrapher, and book designer. He attained prominence as an illustrator and commercial artist, and he brought to the designing of type and books so ...
. The Dozenal Society of Great Britain (DSGB) proposed symbols . This notation, derived from Arabic digits by 180° rotation, was introduced by
Isaac Pitman Sir Isaac Pitman (4 January 1813 – 22 January 1897) was a teacher of the :English language who developed the most widely used system of shorthand, known now as Pitman shorthand. He first proposed this in ''Stenographic Soundhand'' in 183 ...
. In March 2013, a proposal was submitted to include the digit forms for ten and eleven propagated by the Dozenal Societies in the
Unicode Standard Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, whic ...
. Of these, the British/Pitman forms were accepted for encoding as characters at code points and . They were included in
Unicode 8.0 Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, whi ...
(2015). After the Pitman digits were added to Unicode, the DSA took a vote and then began publishing content using the Pitman digits instead. They still use the letters X and E in
ASCII text ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...
. As the Unicode characters are poorly supported, this page uses and . Other proposals are more creative or aesthetic; for example, many do not use any
Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write Decimal, decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers ...
under the principle of "separate identity."


Base notation

There are also varying proposals of how to distinguish a duodecimal number from a decimal one. They include italicizing duodecimal numbers "''54'' = 64", adding a "Humphrey point" (a
semicolon The semicolon or semi-colon is a symbol commonly used as orthographic punctuation. In the English language, a semicolon is most commonly used to link (in a single sentence) two independent clauses that are closely related in thought. When a ...
instead of a
decimal point A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
) to duodecimal numbers "54;6 = 64.5", or some combination of the two. Others use subscript or affixed labels to indicate the base, allowing for more than decimal and duodecimal to be represented (for single letters 'z' from "dozenal" is used as 'd' would mean decimal) such as "54z = 64d," "5412 = 6410" or "doz 54 = dec 64."


Pronunciation

The Dozenal Society of America suggested the pronunciation of ten and eleven as "dek" and "el". For the names of powers of twelve there are two prominent systems.


Duodecimal numbers

In this system, the prefix ''e''- is added for fractions. Multiple digits in this series are pronounced differently: 12 is "do two"; 30 is "three do"; 100 is "gro"; 9 is "el gro dek do nine"; 86 is "el gro eight do six"; 8,15 is "eight gro el do el, one gro five do dek" ABA is "dek gro el do dek" BBB is "el gro el do el" and 0.06 is "six egro" and so on.


Systematic Dozenal Nomenclature (SDN)

This system uses "-qua" ending for the positive powers of 12 and "-cia" ending for the negative powers of 12, and an extension of the IUPAC
systematic element name A systematic element name is the temporary name assigned to an unknown or recently synthesized chemical element. A systematic symbol is also derived from this name. In chemistry, a transuranic element receives a permanent name and symbol only af ...
s (with syllables dec and lev for the two extra digits needed for duodecimal) to express which power is meant.


Advocacy and "dozenalism"

William James Sidis William James Sidis (; April 1, 1898 – July 17, 1944) was an American child prodigy with exceptional mathematical and linguistic skills. He is notable for his 1920 book ''The Animate and the Inanimate'', in which he speculates about the origi ...
used 12 as the base for his constructed language Vendergood in 1906, noting it being the smallest number with four factors and its prevalence in commerce. The case for the duodecimal system was put forth at length in Frank Emerson Andrews' 1935 book ''New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics''. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized ''either'' by the adoption of ten-based weights and measure ''or'' by the adoption of the duodecimal number system. Both the Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the base-twelve system. They use the word "dozenal" instead of "duodecimal" to avoid the more overtly base-ten terminology. However, the etymology of "dozenal" itself is also an expression based on base-ten terminology since "dozen" is a direct derivation of the French word ''douzaine'' which is a derivative of the French word for twelve, ''
douze The Douze (/duz/, oc, Dosa) is the right source river of the Midouze, in the Landes, in the southwest of France. It is long. Geography The Douze rises in Armagnac, in the Gers département. It joins the Midou in Mont-de-Marsan to constitute ...
'', descended from Latin ''duodecim''. Since at least as far back as 1945 some members of the Dozenal Society of America and Dozenal Society of Great Britain have suggested that a more apt word would be "uncial". Uncial is a derivation of the Latin word ''uncia'', meaning "one-twelfth", and also the base-twelve analogue of the Latin word ''decima'', meaning "one-tenth". Mathematician and mental calculator
Alexander Craig Aitken Alexander Craig "Alec" Aitken (1 April 1895 – 3 November 1967) was one of New Zealand's most eminent mathematicians. In a 1935 paper he introduced the concept of generalized least squares, along with now standard vector/matrix notation fo ...
was an outspoken advocate of duodecimal:


In media

In "Little Twelvetoes", American television series ''
Schoolhouse Rock! ''Schoolhouse Rock!'' is an American interstitial programming series of animated musical educational short films (and later, videos) that aired during the Saturday morning children's programming block on the U.S. television network ABC. The theme ...
'' portrayed an alien being using base-twelve arithmetic, using "dek" and "el" as names for ten and eleven, and Andrews' script-X and script-E for the digit symbols.


Duodecimal systems of measurements

Systems of measurement A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce. Systems of measurement i ...
proposed by dozenalists include: * Tom Pendlebury's TGM system * Takashi Suga's Universal Unit System * John Volan's Primel system


Comparison to other number systems

The Dozenal Society of America argues that if a base is too small, significantly longer expansions are needed for numbers; and if a base is too large, one must memorise a large multiplication table to perform arithmetic. Thus it presumes that "a number base will need to be between about 7 or 8 through about 16, possibly including 18 and 20". The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12, of which 2 and 3 are
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 ...
. It is the smallest number to have six factors, the largest number to have at least half of the numbers below it as divisors, and is not much larger than 10. (The numbers 18 and 20 also have six factors, but are much larger.) The decimal system has only four factors, which are 1, 2, 5, and 10, of which 2 and 5 are prime.
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 ...
(base 6) shares the prime factors 2 and 3 with duodecimal, but like decimal it has only four factors (1, 2, 3, and 6) instead of six, and it is below the DSA's stated threshold.
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, ...
(base 8) has four factors, 1, 2, 4 and 8, but has only one prime factor (2).
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 ...
(base 16) adds 16 as a fifth factor, but still no additional prime. Trigesimal (base 30) is the smallest system that has three different prime factors (all of the three smallest primes: 2, 3 and 5) and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30).
Sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
— which the ancient
Sumer Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of c ...
ians and
Babylonia Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. ...
ns among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors. The smallest system that has four different prime factors is base 210 and the pattern follows the
primorial In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...
s. However, these are very large bases. In all base systems, there are similarities to the representation of multiples of numbers which are one less than or one more than the base.


Conversion tables to and from decimal

To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any duodecimal number between 0;01 and ,; to decimal, or any decimal number between 0.01 and 999,999.99 to duodecimal. To use them, the given number must first be decomposed into a sum of numbers with only one significant digit each. For example: :123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 102,304.05), these are, of course, left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). Then the digit conversion tables can be used to obtain the equivalent value in the target base for each digit. If the given number is in duodecimal and the target base is decimal, we get: :(duodecimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0;7 + 0;08 = (decimal) 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.58333333333... + 0.05555555555... Now, because the summands are already converted to base ten, the usual decimal arithmetic is used to perform the addition and recompose the number, arriving at the conversion result: Duodecimal -----> Decimal 100,000 = 248,832 20,000 = 41,472 3,000 = 5,184 400 = 576 50 = 60 + 6 = + 6 0;7 = 0.58333333333... 0;08 = 0.05555555555... -------------------------------------------- 123,456;78 = 296,130.63888888888... That is, (duodecimal) 123,456.78 equals (decimal) 296,130.63 ≈ 296,130.64 If the given number is in decimal and the target base is duodecimal, the method is basically same. Using the digit conversion tables: (decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (duodecimal) 49,54 + ,68 + 1,80 + 294 + 42 + 6 + 0;84972497249724972497... + 0;062... However, in order to do this sum and recompose the number, now the addition tables for the duodecimal system have to be used, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in duodecimal as well. In decimal, 6 + 6 equals 12, but in duodecimal it equals 10; so, if using decimal arithmetic with duodecimal numbers one would arrive at an incorrect result. Doing the arithmetic properly in duodecimal, one gets the result: Decimal -----> Duodecimal 100,000 = 49,54 20,000 = ,68 3,000 = 1,80 400 = 294 50 = 42 + 6 = + 6 0.7 = 0;84972497249724972497... 0.08 = 0;062... -------------------------------------------------------- 123,456.78 = 5,540;943... That is, (decimal) 123,456.78 equals (duodecimal) 5,540;9... ≈ 5,540;94


Duodecimal to decimal digit conversion


Decimal to duodecimal digit conversion


Divisibility rules

(In this section, all numbers are written with duodecimal) This section is about the
divisibility rule A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any rad ...
s in duodecimal. ;1 Any integer is divisible by 1. ;2 If a number is divisible by 2 then the unit digit of that number will be 0, 2, 4, 6, 8 or . ;3 If a number is divisible by 3 then the unit digit of that number will be 0, 3, 6 or 9. ;4 If a number is divisible by 4 then the unit digit of that number will be 0, 4 or 8. ;5 To test for divisibility by 5, double the units digit and subtract the result from the number formed by the rest of the digits. If the result is divisible by 5 then the given number is divisible by 5. This rule comes from 21 (5^2). Examples:
13     rule => , 1-2\times3, =5, which is divisible by 5.
25   rule => , 2\texttt B\texttt A-2\times5, = 2\texttt B0(5\times70), which is divisible by 5 (or apply the rule on 20). OR To test for divisibility by 5, subtract the units digit and triple of the result to the number formed by the rest of the digits. If the result is divisible by 5 then the given number is divisible by 5. This rule comes from 13 (5\times3). Examples:
13     rule => , 3-3\times1, =0, which is divisible by 5.
25   rule => , 5-3\times2\texttt B\texttt A, =8\texttt B1(5\times195), which is divisible by 5 (or apply the rule on 81). OR Form the alternating sum of blocks of two from right to left. If the result is divisible by 5 then the given number is divisible by 5. This rule comes from 101, since 101=5\times25; thus, this rule can be also tested for the divisibility by 25. Example: 97,374,627 => 27-46+37-97=-7\texttt B, which is divisible by 5. ;6 If a number is divisible by 6 then the unit digit of that number will be 0 or 6. ;7 To test for divisibility by 7, triple the units digit and add the result to the number formed by the rest of the digits. If the result is divisible by 7 then the given number is divisible by 7. This rule comes from 2 (7\times5) Examples:
12     rule => , 3\times2+1, =7, which is divisible by 7.
271    rule => , 3\times\texttt B+271, =29\texttt A(7\times4\texttt A), which is divisible by 7 (or apply the rule on 29). OR To test for divisibility by 7, subtract the units digit and double the result from the number formed by the rest of the digits. If the result is divisible by 7 then the given number is divisible by 7. This rule comes from 12 (7\times2). Examples:
12     rule => , 2-2\times1, =0, which is divisible by 7.
271    rule => , \texttt B-2\times271, =513 (7\times89), which is divisible by 7 (or apply the rule on 513). OR To test for divisibility by 7, quadruple the units digit and subtract the result from the number formed by the rest of the digits. If the result is divisible by 7 then the given number is divisible by 7. This rule comes from 41 (7^2). Examples:
12     rule => , 4\times2-1, =7, which is divisible by 7.
271    rule => , 4\times\texttt B-271, =235(7\times3\texttt B), which is divisible by 7 (or apply the rule on 235). OR Form the alternating sum of blocks of three from right to left. If the result is divisible by 7 then the given number is divisible by 7. This rule comes from 1001, since 1001=7\times11\times17, thus this rule can be also tested for the divisibility by 11 and 17. Example: 386,967,443 => 443-967+386=-168, which is divisible by 7. ;8 If the 2-digit number formed by the last 2 digits of the given number is divisible by 8 then the given number is divisible by 8. Example: 148, 4120 rule => since 48(8*7) divisible by 8, then 148 is divisible by 8. rule => since 20(8*3) divisible by 8, then 4120 is divisible by 8. ;9 If the 2-digit number formed by the last 2 digits of the given number is divisible by 9 then the given number is divisible by 9. Example: 7423, 8330 rule => since 23(9*3) divisible by 9, then 7423 is divisible by 9. rule => since 30(9*4) divisible by 9, then 8330 is divisible by 9. ; If the number is divisible by 2 and 5 then the number is divisible by . ; If the sum of the digits of a number is divisible by then the number is divisible by (the equivalent of
casting out nines Casting out nines is any of three arithmetical procedures: *Adding the decimal digits of a positive whole number, while optionally ignoring any 9s or digits which sum to a multiple of 9. The result of this procedure is a number which is smaller th ...
in decimal). Example: 29, 6113 rule => 2+9 = which is divisible by , then 29 is divisible by . rule => 6+1++1+3 = 1 which is divisible by , then 6113 is divisible by . ;10 If a number is divisible by 10 then the unit digit of that number will be 0. ;11 Sum the alternate digits and subtract the sums. If the result is divisible by 11 the number is divisible by 11 (the equivalent of divisibility by eleven in decimal). Example: 66, 9427 rule => , 6-6, = 0 which is divisible by 11, then 66 is divisible by 11. rule => , (9+2)-(4+7), = , -, = 0 which is divisible by 11, then 9427 is divisible by 11. ;12 If the number is divisible by 2 and 7 then the number is divisible by 12. ;13 If the number is divisible by 3 and 5 then the number is divisible by 13. ;14 If the 2-digit number formed by the last 2 digits of the given number is divisible by 14 then the given number is divisible by 14. Example: 1468, 7394 rule => since 68(14*5) divisible by 14, then 1468 is divisible by 14. rule => since 94(14*7) divisible by 14, then 7394 is divisible by 14.


Fractions and irrational numbers


Fractions

Duodecimal
fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
may be simple: * = 0;6 * = 0;4 * = 0;3 * = 0;2 * = 0;16 * = 0;14 * = 0;1 (this is a twelfth, is a tenth) * = 0;09 (this is a sixteenth, is a fourteenth) or complicated: * = 0;... recurring (rounded to 0;24) * = 0;... recurring (rounded to 0;187) * = 0;1... recurring (rounded to 0;125) * = 0;... recurring (rounded to 0;111) * = 0;... recurring (rounded to 0;01) * = 0;0... recurring (rounded to 0;03) * = 0;0... recurring (rounded to 0;097) As explained in
recurring decimal A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...
s, whenever an
irreducible fraction An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). ...
is written in
radix point A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
notation in any base, the fraction can be expressed exactly (terminates) if and only if all the
prime factor 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 ...
s of its denominator are also prime factors of the base. Because , in the decimal system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate:  = ,  =  and  =  can be expressed exactly as 0.125, 0.05 and 0.002 respectively. and , however, recur (0.333... and 0.142857142857...). Because , in the duodecimal system, is exact; and recur because they include 5 as a factor; is exact; and recurs, just as it does in decimal. The number of denominators which give terminating fractions within a given number of digits, say ''n'', in a base ''b'' is the number of factors (divisors) of ''bn'', the ''n''th power of the base ''b'' (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of ''bn'' is given using its prime factorization. For decimal, . The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together, so the number of factors of is . For example, the number 8 is a factor of 103 (1000), so 1/8 and other fractions with a denominator of 8 cannot require more than 3 fractional decimal digits to terminate. 5/8 = 0.62510 For duodecimal, . This has divisors. The sample denominator of 8 is a factor of a gross in decimal), so eighths cannot need more than two duodecimal fractional places to terminate. Because both ten and twelve have two unique prime factors, the number of divisors of for grows quadratically with the exponent ''n'' (in other words, of the order of ''n''2).


Recurring digits

The Dozenal Society of America argues that factors of 3 are more commonly encountered in real-life
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
problems than factors of 5. Thus, in practical applications, the nuisance of repeating decimals is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations. However, when recurring fractions ''do'' occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because 12 (twelve) is between two
prime number 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 ...
s, 11 (eleven) and 13 (thirteen), whereas ten is adjacent to the
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
9. Nonetheless, having a shorter or longer period doesn't help the main inconvenience that one does not get a finite representation for such fractions in the given base (so
rounding Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression with . Rounding is often done to obta ...
, which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor 3 in its factorization, whereas only one out of every five contains the prime factor 5. All other prime factors, except 2, are not shared by either ten or twelve, so they do not influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base). Also, the prime factor 2 appears twice in the factorization of twelve, whereas only once in the factorization of ten; which means that most fractions whose denominators are
powers of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
will have a shorter, more convenient terminating representation in duodecimal than in decimal: * 1/(22) = 0.2510 = 0.312 * 1/(23) = 0.12510 = 0.1612 * 1/(24) = 0.062510 = 0.0912 * 1/(25) = 0.0312510 = 0.04612 The duodecimal period length of 1/''n'' are (in decimal) :0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, ... The duodecimal period length of 1/(''n''th prime) are (in decimal) :0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280, ... Smallest prime with duodecimal period ''n'' are (in decimal) :11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, ...


Irrational numbers

The representations of
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
s in any positional number system (including decimal and duodecimal) neither terminate nor repeat. The following table gives the first digits for some important algebraic and transcendental numbers in both decimal and duodecimal.


See also

*
Vigesimal vigesimal () or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). '' Vigesimal'' is derived from the Latin adjective '' vicesimus'', meaning 'twentieth'. Places In ...
(base 20) *
Sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
(base 60)


References


Further reading

* * {{cite web , title=Computer Arithmetic , at=The Early Days of Hexadecimal , author-first=John J. G. , author-last=Savard , date=2018 , orig-year=2005 , work=quadibloc , url=http://www.quadibloc.com/comp/cp02.htm , access-date=2018-07-16 , url-status=live , archive-url=https://web.archive.org/web/20180716102439/http://www.quadibloc.com/comp/cp02.htm , archive-date=2018-07-16 (NB. Also has information on duodecimal representations.)


External links


Dozenal Society of America

Dozenal Society of Great Britain



Comprehensive Synopsis of Dozenal and Transdecimal Symbologies
Positional numeral systems 12 (number)