Duodecimal


The duodecimal system, also known as base twelve or dozenal, is a positional numeral system using twelve as its base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 units; in the decimal system, this number is instead written as "12" meaning 1 ten and 2 units, and the string "10" means ten. In duodecimal, "100" means twelve squared, "1,000" means twelve cubed, and "0.1" means a twelfth.
Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses and, as in hexadecimal, which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,,, and finally 10. The Dozenal Societies of America and Great Britain use turned digits in their published material: 2 for ten and 3 for eleven.
The number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range, and the smallest abundant number. All multiples of reciprocals of 3-smooth numbers have a terminating representation in duodecimal. In particular, , , , , and 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 or hexadecimal. Sexagesimal does even better in this respect, but at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.

Origin

Georges Ifrah speculatively traced the origin of the duodecimal system to a system of finger counting based on the knuckle bones of the four larger fingers. Using the thumb as a pointer, it is possible to count to 12 by touching each finger bone, starting with the farthest bone on the fifth finger, and counting on. In this system, one hand counts repeatedly to 12, while the other displays the number of iterations, until five dozens, i.e. the 60, are full. This system is still in use in many regions of Asia.
Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu, Piti, and the Nimbia dialect of Gwandara; and the Chepang language of Nepal are known to use duodecimal numerals.
Germanic languages have special words for 11 and 12, such as eleven and twelve in English. They come from Proto-Germanic *ainlif and *twalif, suggesting a decimal rather than duodecimal origin. However, Old Norse used a hybrid decimal–duodecimal counting system, with its words for "one hundred and eighty" meaning 200 and "two hundred" meaning 240. In the British Isles, this style of counting survived well into the Middle Ages as the long hundred.
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. Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches or 24 Solar terms. There are 12 inches in an imperial foot, 12 troy ounces in a troy pound, 24 hours in a day; many other items are counted by the dozen, gross, or great gross. The Romans used a fraction system based on 12, including the uncia, which became both the English words ounce and inch. Historically, many parts of western Europe used a mixed vigesimal–duodecimal currency system of pounds, shillings, and pence, with 20 shillings to a pound and 12 pence to a shilling, originally established by Charlemagne in the 780s.

Notations and pronunciations

In a positional numeral system of base n, each of the first n natural numbers is given a distinct numeral symbol, and then n is denoted "10", meaning 1 times n plus 0 units. For duodecimal, the standard numeral symbols for 0–9 are typically preserved for zero through nine, but there are numerous proposals for how to write the numerals representing "ten" and "eleven". More radical proposals do not use any Arabic numerals under the principle of "separate identity."
Pronunciation of duodecimal numbers also has no standard, but various systems have been proposed.

Transdecimal symbols

Several authors have proposed using letters of the alphabet for the transdecimal symbols. Latin letters such as or are convenient because they are widely accessible, and for instance can be typed on typewriters. However, when mixed with ordinary prose, they might be confused for letters. As an alternative, Greek letters such as could be used instead. Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his 1935 book New Numbers , along with italic numerals –.
Edna Kramer in her 1951 book The Main Stream of Mathematics used ⟨⟩ and ⟨⟩. The symbols were chosen because they were available on some typewriters; they are also on push-button telephones. This notation was used in publications of the Dozenal Society of America from 1974 to 2008.
From 2008 to 2015, the DSA used, the symbols devised by William Addison Dwiggins.
The Dozenal Society of Great Britain proposed symbols. This notation, derived from Arabic digits by 180° rotation, was introduced by Isaac Pitman in 1857. 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. Of these, the British/Pitman forms were accepted for encoding as characters at code points and. They were included in Unicode 8.0.
After the Pitman digits were added to Unicode, the DSA took a vote and then began publishing PDF content using the Pitman digits instead, but continues to use the letters X and E on its webpage.

Base notation

There are also varying proposals of how to distinguish a duodecimal number from a decimal one. The most common method used in mainstream mathematics sources comparing various number bases uses a subscript "10" or "12", e.g. " = ". To avoid ambiguity about the meaning of the subscript 10, the subscripts might be spelled out, " = ". In 2015 the Dozenal Society of America adopted the more compact single-letter abbreviation z for dozenal and d for decimal, " = ".
Other proposed methods include italicizing duodecimal numbers "54 = 64", adding a "Humphrey point" to duodecimal numbers "54;6 = 64.5", prefixing duodecimal numbers by an asterisk "*54 = 64", or some combination of these. The Dozenal Society of Great Britain uses an asterisk prefix for duodecimal whole numbers, and a Humphrey point for other duodecimal numbers.

Pronunciation

The Dozenal Society of America suggested ten and eleven should be pronounced as "dek" and "el", respectively.
Terms for some powers of twelve already exist in English: The number twelve is also called a dozen. Twelve squared is called a gross. Twelve cubed is called a great gross.

Advocacy and "dozenalism"

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 adoption of the duodecimal system.
Mathematician and mental calculator Alexander Craig Aitken was an outspoken advocate of duodecimal:

In media

In "Little Twelvetoes," an episode of the American educational television series Schoolhouse Rock!, a farmer encounters an alien being with a total of twelve fingers and twelve toes who uses duodecimal arithmetic. The alien uses "dek" and "el" as names for ten and eleven, and Andrews' script-X and script-E for the digit symbols.

Duodecimal systems of measurements

proposed by dozenalists include Tom Pendlebury's TGM system, Takashi Suga's Universal Unit System, and 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; 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. 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 only slightly larger than 10. Ten, in contrast, only has four factors, which are 1, 2, 5, and 10, of which 2 and 5 are prime. Six shares the prime factors 2 and 3 with twelve; however, like ten, six only has four factors instead of six. Its corresponding base, senary, is below the DSA's stated threshold.
Eight and sixteen only have 2 as a prime factor. Therefore, in octal and hexadecimal, the only terminating fractions are those whose denominator is a power of two.
Thirty is the smallest number that has three different prime factors, and it has eight factors in total. Sexagesimal was actually used by the ancient Sumerians and Babylonians, among others; its base, sixty, adds the four convenient factors 4, 12, 20, and 60 to 30 but no new prime factors. The smallest number that has four different prime factors is 210; the pattern follows the primorials. However, these numbers are quite large to use as bases, and are far beyond the DSA's stated threshold.
In all base systems, there are similarities to the representation of multiples of numbers that are one less than or one more than the base.In the following multiplication table, numerals are written in duodecimal. For example, "10" means twelve, and "12" means fourteen.
×12345678910
112345678910
224681012141618120
3369101316192023262930
44810141820242830343840
551318212623439424750
661016202630364046505660
7712192423641485356570
881420283440485460687480
991623303946536069768390
1826344250568768492A0
129384756657483921B0
10102030405060708090A0B0100