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{{distinguish|Dewey Decimal Classification|Duodecimo}}{{Special characters}}{{Numeral systems}}{{Very long|date=March 2018}}The duodecimal system (also known as base 12 or dozenal) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the base ten numerical system) is instead written as "10" in duodecimal (meaning "1 dozen and 0 units", instead of "1 ten and 0 units"), whereas the digit string "12" means "1 dozen and 2 units" (i.e. the same number that in decimal is written as "14"). Similarly, in duodecimal "100" means "1 gross", "1000" means "1 great gross", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth").The number twelve, a superior highly composite number, is the smallest number with four non-trivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range, and the smallest abundant number. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, and not 3, 4, or 6), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system.WEB
, Why We Should Switch To A Base-12 Counting System
, George Dvorsky
, 2013-01-18
, 2013-12-21
, bot: unknown
,weblink" title="">weblink
, 2013-01-21
, Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers (such as 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, ...) have a terminating representation in duodecimal. In particular, the five most elementary fractions ({{frac||1|2}}, {{frac||1|3}}, {{frac||2|3}}, {{frac||1|4}} and {{frac||3|4}}) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (because it is the least common multiple of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary, octal and hexadecimal systems. Although the trigesimal and sexagesimal systems (where the reciprocals of all 5-smooth numbers terminate) do even better in this respect, this is at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.


In this section, numerals are based on decimal places. For example, 10 means ten, 12 means twelve.
Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Gure-Kahugu), Piti, and the Nimbia dialect of Gwandara;CONFERENCE, Decimal vs. Duodecimal: An interaction between two systems of numeration, Matsushita, Shuji, 2nd Meeting of the AFLANG, October 1998, Tokyo, 1998,weblinkweblink" title="">weblink 2008-10-05, 2011-05-29, the Chepang language of NepalBOOK, Les principes de construction du nombre dans les langues tibéto-birmanes, Martine, Mazaudon, La Pluralité, Jacques, François, 2002, 91–119, Peeters, Leuven, 90-429-1295-2,weblink and the Maldivian language (Dhivehi) of the people of the Maldives and Minicoy Island in India are known to use duodecimal numerals. Germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal.BOOK, von Mengden, Ferdinand, 2006, The peculiarities of the Old English numeral system, Medieval English and its Heritage: Structure Meaning and Mechanisms of Change, Nikolaus Ritt, Herbert Schendl, Christiane Dalton-Puffer, Dieter Kastovsky, Peter Lang, Studies in English Medieval Language and Literature, 16, Frankfurt, 125–145, BOOK, von Mengden, Ferdinand, 2010, Cardinal Numerals: Old English from a Cross-Linguistic Perspective, Topics in English Linguistics, 67, Berlin; New York, De Gruyter Mouton, 159–161, Historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, and the Babylonians had twelve hours in a day (although at some point this was changed to 24.) Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in an imperial foot, 12 troy ounces in a troy pound, 12 old British pence in a shilling, 24 (12×2) hours in a day, and many other items counted by the dozen, gross (144, square of 12) or great gross (1728, cube of 12). The Romans used a fraction system based on 12, including the uncia which became both the English words ounce and inch. Pre-decimalisation, Ireland and the United Kingdom used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the pound sterling or Irish pound), and Charlemagne established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.{| class="wikitable" style=text-align:center! colspan="6"|Table of units from a base of 12! Relativevalue! French unitof length! English unitof length! English(Troy) unitof weight! Roman unitof weight! English unitof mass
Units of measurement in France before the French Revolution#Length>piedFoot (unit)>footTroy weight#Units of measurement>poundAncient Roman units of measurement#Weight>libra|
Units of measurement in France before the French Revolution#Length>pouce|inchTroy weight#Units of measurement>ounceAncient Roman units of measurement#Weight>unciaslug (unit)#Similar units>slinch
Units of measurement in France before the French Revolution#Length>ligneLine (unit)>lineApothecaries' system#English-speaking countries>scruplesscrupulum>scrupulaslug (unit)>slug
Point (typography)#Truchet>pointPoint (typography)#Truchet>pointsiliqua>seed|siliqua |
The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (three on each of four fingers).JOURNAL, Pittman, Richard, 1990, Origin of Mesopotamian duodecimal and sexagesimal counting systems, Philippine Journal of Linguistics, 21, 1, 97, WEB, ja, ヒマラヤの満月と十二進法, The Full Moon in the Himalayas and the Duodecimal System, Nishikawa, Yoshiaki, 2002,weblink 2008-03-24, yes,weblink" title="">weblink March 29, 2008, It is possible to count to 12 with the thumb acting as a pointer, touching each finger bone in turn. A traditional finger counting 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.BOOK
, Ifrah
, Georges
, Georges Ifrah
, The Universal History of Numbers: From prehistory to the invention of the computer
, John Wiley and Sons
, 2000
, 0-471-39340-1
, Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk.BOOK, Macey, Samuel L., The Dynamics of Progress: Time, Method, and Measure, 1989, University of Georgia Press, Atlanta, Georgia, 978-0-8203-3796-8, 92,weblink

Notations and pronunciations

Transdecimal symbols

In a duodecimal place system twelve is written as 10, but there are numerous proposals for how to write ten and eleven.JOURNAL, De Vlieger, Michael, Symbology Overview, The Duodecimal Bulletin, 4X [58], 2, 2010,weblink The simplified notations use only basic and easy to access letters such as A and B (as in the hexadecimal and vigesimal), T and E (initials of Ten and Eleven), X and Z. Some employ Greek letters such as δ (standing for Greek δέκα 'ten') and ε (for Greek ένδεκα 'eleven'), or τ and ε. Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his book New Numbers an X (from the Roman numeral for ten) and a script E (ℰ, {{U+|2130}}).BOOK, Frank Emerson, Andrews, New Numbers: How Acceptance of a Duodecimal (12) Base Would Simplify Mathematics, 1935, 52,weblink {{Symb|(File:Dozenal gb 10.svg) (File:Dozenal gb 11.svg)(File:Duodecimal-digit-ten-Dozenal-Society-of-America.svg) (File:Dozenal us 11.svg)}}The Dozenal Society of Great Britain proposes a rotated digit two 2 for ten and a reversed or rotated digit three 3 for eleven. This notation was introduced by Sir Isaac Pitman.Pitman, Isaac (ed.): A triple (twelve gross) Gems of Wisdom. London 1860JOURNAL, Pitman, Isaac, A Reckoning Reform [reprint from 1857], The Duodecimal Bulletin, 3, 2, 1947,weblink Until 2015, the Dozenal Society of America (DSA) used (File:Dozenal us 10.svg|15x15px) and (File:Dozenal us 11.svg|15x15px), the symbols devised by William Addison Dwiggins.JOURNAL, Mo for Megro, The Duodecimal Bulletin, 1, 1, 1945,weblink After the Pitman digits (32) were added to Unicode the DSA took a vote and then began publishing content using the Pitman digits instead.WEB,weblink What should the DSA do about transdecimal characters?, The Dozenal Society of America, 2018-01-01, JOURNAL, Volan, John, July 2015, Base Annotation Schemes,weblink Duodecomal Bulletin, 62, They still use the letters X and E as the equivalent in ASCII text.Other proposals are more creative or aesthetic, for example, Edna Kramer in her 1951 book The Main Stream of Mathematics used a six-pointed asterisk (sextile) ⚹ for ten and a hash (or octothorpe) # for eleven. The symbols were chosen because they are available in typewriters and already present in telephone dials. This notation was used in publications of the Dozenal Society of America in the period 1974–2008.JOURNAL, Annual Meeting of 1973 and Meeting of the Board, The Duodecimal Bulletin, 25 [29], 1, 1974,weblink JOURNAL, De Vlieger, Michael, Going Classic, The Duodecimal Bulletin, 49 [57], 2, 2008,weblink Many don't use any Arabic numerals under the principle of "separate identity."In March 2013, a proposal was submitted to include the digit forms for ten and eleven propagated by the Dozenal Societies of Great Britain and America in the Unicode Standard.WEB,weblink Proposal to encode Duodecimal Digit Forms in the UCS, 2013-03-30, ISO/IEC JTC1/SC2/WG2, Document N4399, 2016-05-30, Karl Pentzlin, Of these, the British/Pitman forms were accepted for encoding as characters at code points {{unichar|218A|TURNED DIGIT TWO}} and {{unichar|218B|TURNED DIGIT THREE}}. They were included in the Unicode 8.0 release in June 2015.WEB,weblink The Unicode Standard, Version 8.0: Number Forms, Unicode Consortium, 2016-05-30, WEB
, The Unicode Standard 8.0
, 2014-07-18
, Fonts which support these characters include EB Garamond, Everson Mono, and Squarish Sans CT. As these Unicode symbols are not rendering in many browsers, this article uses {{unichar|2D52|TIFINAGH LETTER YAP}} and {{unichar|0190|LATIN CAPITAL LETTER OPEN E}}, or physically rotated '2' and '3' glyphs.The turned digits two and three are available in LaTeX as textturntwo and textturnthree.WEB,weblink The Comprehensive LATEX Symbol List, 2009, 2016-05-30, Scott Pakin,

{{anchor|Humphrey point}}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 ";" instead of a decimal point ".") 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." Programming languages limited to ASCII could use a prefix, similar to 0x used for hexadecimal, perhaps 0z but there is no standard.


The Dozenal Society of America suggests the pronunciation of ten and eleven as "dek" and "el", each order has its own name and the prefix e- is added for fractions.JOURNAL, Zirkel, Gene, How Do You Pronounce Dozenals?volume=4E [59]date=2010,weblink The symbol corresponding to the decimal point or decimal comma, separating the whole number part from the fractional part, is the semicolon ";". The overall system is:{| class="wikitable"! Duodecimal||Name||Decimal||Duodecimal fraction||Name
10do|pronounced 'doh'}}120;1edo
100gro|pronounced 'groh'}}1440;01|egro
1,000mo|pronounced 'moh'}}1,7280;001|emo
Multiple digits in this are pronounced differently. 12 is "one do two", 30 is "three do", 100 is "one gro", BA9 (ET9) is "el gro dek do nine", B8,65A,300 (E8,65T,300) is "el do eight bi-mo, six gro five do dek mo, three gro", and so on.

Advocacy and "dozenalism"

William James Sidis used 12 as the base for his constructed language Vendergood in 1906, noting it being the smallest number with four factors and the prevalence in commerceThe Prodigy (Biography of WJS) pg [42].The case for the duodecimal system was put forth at length in F. 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.File:Keys in dozenal clock.svg|thumb|A duodecimal clockface as in the logo of the Dozenal Society of America, here used to denote musical keys ]]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. It should be noted that the etymology of 'dozenal' is itself 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, (:wikt:douze|douze) which is related to the old French word 'doze' from Latin 'duodecim'. Since at least as far back as 1945 some members of the Dozenal Society of America and Duodecimal Society of Great Britain have suggested that a more apt word would be 'uncial'. Uncial is a derivation of the Latin word 'one-twelfth' which is 'uncia' and also the base-twelve analogue of the Latin word 'one-tenth' which is 'decima'. In the same manner as decimal comes from the Latin word for one-tenth decima, (Latin for ten was decem), the direct analogue for a base-twelve system is uncial.William S. Crosby; "The Uncial Jottings of a Harried Infantryman", The Duodecimal Bulletin, Vol 1 Issue 2, June 1945, Page 9.Mathematician and mental calculator Alexander Craig Aitken was an outspoken advocate of duodecimal:In Jorge Luis Borges' short story "Tlön, Uqbar, Orbis Tertius" Herbert Ashe, a melancholy English engineer, working for the Southern Argentine Railway company, is converting a duodecimal number system to a hexadecimal system. He leaves behind on his death in 1937 a manuscript Orbis Tertius that posthumously identifies him as one of the anonymous authors of the encyclopaedia of Tlön.In Leo Frankowski's Conrad Stargard novels, Conrad introduces a duodecimal system of arithmetic at the suggestion of a merchant, who is accustomed to buying and selling goods in dozens and grosses, rather than tens or hundreds. He then invents an entire system of weights and measures in base twelve, including a clock with twelve hours in a day, rather than twenty-four hours.{{citation needed|date=May 2016}}In Lee Carroll's Kryon: Alchemy of the Human Spirit, a chapter is dedicated to the advantages of the duodecimal system. The duodecimal system is supposedly suggested by Kryon (a fictional entity believed in by New Age circles) for all-round use, aiming at better and more natural representation of nature of the Universe through mathematics. An individual article "Mathematica" by James D. Watt (included in the above publication) exposes a few of the unusual symmetry connections between the duodecimal system and the golden ratio, as well as provides numerous number symmetry-based arguments for the universal nature of the base-12 number system.BOOK, Lee, Carroll, Kryon—Alchemy of the Human Spirit,weblink 1995, The Kryon Writings, Inc., 0-9636304-8-2, In "Little Twelvetoes", American television series Schoolhouse Rock! portrayed an alien child using base-twelve arithmetic, using "dek", "el" and "doh" as names for ten, eleven and twelve, and Andrews' script-X and script-E for the digit symbols.WEB,weblink" title="">weblink SchoolhouseRock - Little Twelvetoes, 6 February 2010,,

Duodecimal metric systems

Systems of measurement proposed by dozenalists include:
  • Tom Pendlebury's TGM systemWEB, Pendlebury, Tom, TGM. A coherent dozenal metrology based on Time, Gravity and Mass,weblink May 2011, The Dozenal Society of Great Britain, WEB, Goodman, Donald, Manual of the Dozenal System,weblink Dozenal Society of America, 27 April 2018,
  • Takashi Suga's Universal Unit SystemWEB, Suga, Takashi, Proposal for the Universal Unit System,weblink 2002,

Comparison to other number systems

The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12, of which 2 and 3 are prime. The decimal system has only four factors, which are 1, 2, 5, and 10, of which 2 and 5 are prime. Vigesimal (base 20) adds two factors to those of ten, namely 4 and 20, but no additional prime factor. Although twenty has 6 factors, 2 of them prime, similarly to twelve, it is also a much larger base, and so the digit set and the multiplication table are much larger. Binary has only two factors, 1 and 2, the latter being prime. Hexadecimal (base 16) has five factors, adding 4, 8 and 16 to those of 2, but 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—which the ancient Sumerians and Babylonians 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 primorials. In all base systems, there are similarities to the representation of multiples of numbers which are one less than the base.{{Clear}}{|class="wikitable|+ Duodecimal multiplication table!×!!0!!1!!2!!3!!4!!5!!6!!7!!8!!9!!ⵒ!!Ɛ!0

Conversion tables to and from decimal

To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under divisibility rules in duodecimal.
Any integer is divisible by 1.
If a number is divisible by 2 then the unit digit of that number will be 0, 2, 4, 6, 8 or âµ’.
If a number is divisible by 3 then the unit digit of that number will be 0, 3, 6 or 9.
If a number is divisible by 4 then the unit digit of that number will be 0, 4 or 8.
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*5)Examples: 13     rule => |1-2*3| = 5 which is divisible by 5.2Ɛⵒ5   rule => |2Ɛⵒ-2*5| = 2Ɛ0(5*70) which is divisible by 5(or apply the rule on 2Ɛ0).ORTo 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*3)Examples: 13     rule => |3-3*1| = 0 which is divisible by 5.2Ɛⵒ5   rule => |5-3*2Ɛⵒ| = 8Ɛ1(5*195) which is divisible by 5(or apply the rule on 8Ɛ1).ORForm 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*25, thus this rule can be also tested for the divisibility by 25.Example:97,374,627 => 27-46+37-97 = -7Ɛ which is divisible by 5.
If a number is divisible by 6 then the unit digit of that number will be 0 or 6.
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 2Ɛ(7*5)Examples:12     rule => |3*2+1| = 7 which is divisible by 7.271Ɛ    rule => |3*Ɛ+271| = 29âµ’(7*4âµ’) which is divisible by 7(or apply the rule on 29âµ’).ORTo 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*2)Examples:12     rule => |2-2*1| = 0 which is divisible by 7.271Ɛ    rule => |Ɛ-2*271| = 513(7*89) which is divisible by 7(or apply the rule on 513).ORTo test for divisibility by 7, 4 times 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*7)Examples:12     rule => |4*2-1| = 7 which is divisible by 7.271Ɛ    rule => |4*Ɛ-271| = 235(7*3Ɛ) which is divisible by 7(or apply the rule on 235).ORForm 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*11*17, 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.
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: 1Ɛ48, 4120
rule => since 48(8*7) divisible by 8, then 1Ɛ48 is divisible by 8.
rule => since 20(8*3) divisible by 8, then 4120 is divisible by 8.

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 âµ’.
If the sum of the digits of a number is divisible by Ɛ then the number is divisible by Ɛ (the equivalent of casting out nines in decimal).Example: 29, 61Ɛ13
rule => 2+9 = Ɛ which is divisible by Ɛ, then 29 is divisible by Ɛ.
rule => 6+1+Ɛ+1+3 = 1ⵒ which is divisible by Ɛ, then 61Ɛ13 is divisible by Ɛ.

If a number is divisible by 10 then the unit digit of that number will be 0.
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)| = |âµ’-âµ’| = 0 which is divisible by 11, then 9427 is divisible by 11.

If the number is divisible by 2 and 7 then the number is divisible by 12.
If the number is divisible by 3 and 5 then the number is divisible by 13.
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


Duodecimal fractions may be simple:
  • {{sfrac|2}} = 0.6
  • {{sfrac|3}} = 0.4
  • {{sfrac|4}} = 0.3
  • {{sfrac|6}} = 0.2
  • {{sfrac|8}} = 0.16
  • {{sfrac|9}} = 0.14
  • {{sfrac|10}} = 0.1 (note that this is a twelfth, {{sfrac|âµ’}} is a tenth)
  • {{sfrac|14}} = 0.09 (note that this is a sixteenth, {{sfrac|12}} is a fourteenth)
or complicated:
  • {{sfrac|5}} = 0.249724972497... recurring (rounded to 0.24âµ’)
  • {{sfrac|7}} = 0.186âµ’35186âµ’35... recurring (rounded to 0.187)
  • {{sfrac|âµ’}} = 0.1249724972497... recurring (rounded to 0.125)
  • {{sfrac|Ɛ}} = 0.111111111111... recurring (rounded to 0.111)
  • {{sfrac|11}} = 0.0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ... recurring (rounded to 0.0Ɛ1)
  • {{sfrac|12}} = 0.0âµ’35186âµ’35186... recurring (rounded to 0.0âµ’3)
  • {{sfrac|13}} = 0.0972497249724... recurring (rounded to 0.097)
{|class="wikitable"! Examples in duodecimal! Decimal equivalent
5|8}}) = 0.765|8}}) = 0.625
5|8}}) = 765|8}}) = 90
576|9}} = 76810|9}} = 90
400|9}} = 54576|9}} = 64
| 1âµ’.6 + 7.6 = 26| 22.5 + 7.5 = 30
As explained in recurring decimals, whenever an irreducible fraction is written in radix point notation in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base. Thus, in base-ten (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: {{sfrac|8}} = {{sfrac|(2×2×2)}}, {{sfrac|20}} = {{sfrac|(2×2×5)}} and {{sfrac|500}} = {{sfrac|(2×2×5×5×5)}} can be expressed exactly as 0.125, 0.05 and 0.002 respectively. {{sfrac|3}} and {{sfrac|7}}, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, {{sfrac|8}} is exact; {{sfrac|20}} and {{sfrac|500}} recur because they include 5 as a factor; {{sfrac|3}} is exact; and {{sfrac|7}} 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 nth 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, 10n = 2n * 5n. The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together.Factors of 10n = (n+1)(n+1) = (n+1)2.For example, the number 8 is a factor of 103 (1000), so 1/8 and other fractions with a denominator of 8 can not require more than 3 fractional decimal digits to terminate. 5/8 = 0.625tenFor duodecimal, 12n = 22n * 3n. This has (2n+1)(n+1) divisors. The sample denominator of 8 is a factor of a gross (122 = 144), so eighths can not need more than two duodecimal fractional places to terminate. 5/8 = 0.76twelveBecause both ten and twelve have two unique prime factors, the number of divisors of bn for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of n2).

Recurring digits

The Dozenal Society of America argues that factors of 3 are more commonly encountered in real-life division problems than factors of 5.WEB, Dozenal FAQs, Michael Thomas De Vlieger, The Dozenal Society of America, 30 November 2011,weblink 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 numbers, 11 (eleven) and 13 (thirteen), whereas ten is adjacent to the composite number 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, 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 notinfluence 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 irrational numbers 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.{|class="wikitable" ! Algebraic irrational number! In decimal! In duodecimal
Square root of 22}}, the square root of 2| 1.414213562373...| 1.4Ɛ79170ⵒ07Ɛ8...
Golden ratioφ}} (phi), the golden ratio = tfrac{1+sqrt{5}}{2}| 1.618033988749...| 1.74ƐƐ6772802ⵒ...
! Transcendental number! In decimal! In duodecimal
Piπ}} (pi), the ratio of a circle's circumference to its diameter| 3.141592653589...| 3.184809493Ɛ91...
E (mathematical constant)e}}, the base of the natural logarithm| 2.718281828459...| 2.875236069821...

See also



Further reading

  • WEB, Changing the Base, John J. G., Savard, 2018, 2016, quadibloc,weblink 2018-07-17, no,weblink" title="">weblink 2018-07-17,
  • WEB, Computer Arithmetic, The Early Days of Hexadecimal, John J. G., Savard, 2018, 2005, quadibloc,weblink 2018-07-16, no,weblink" title="">weblink 2018-07-16, (NB. Also has information on duodecimal representations.)

External links

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