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]] The duodecimal (also known as ''' Base -''' or '''dozenal''') system is a Numeral System using Twelve as its base. The number Ten may be written as A, the number Eleven as B, and the number twelve as 10. The number 12 has six factors, which are 1 , 2 , 3 , 4 , 6 , and 12 . It is a more convenient number system for computing fractions compared with the Decimal or Vigesimal systems. The decimal system has only four factors, which are 1 , 2 , 5 , and 10 . ORIGIN In this section, numerals are based on decimal Places . For example, 10 means Ten , 12 means Twelve . Languages based on the duodecimal system are uncommon. Languages in the Nigeria n Middle Belt such as Janji , Kahugu , the Nimbia dialect of Gwandara , Mahl Language of Minicoy and the Chepang language of Nepal are known to use duodecimal numerals. In fiction, J. R. R. Tolkien 's Elvish Languages used the duodecimal. Natural explanations for the choice of the number twelve include the following:
Historically, Unit s of Time in many Civilization s are duodecimal, which may come as a generalization of the use for months. There are twelve signs of the Zodiac . There are twelve European Hour s in a day or night. Traditional Chinese Calendar s, clocks, and compasses are based on the twelve Earthly Branches .
Being a versatile denominator in fractions may explain why we have 12 Inch es in a Foot , 12 ounces in a Troy Pound , 12 Old British Pence in a Shilling , 12 items in a Dozen , 12 dozens in a Gross ( 144 , Square of 12), 12 gross in a Great Gross ( 1728 , Cube of 12), 10 dozens in a Small Gross ( 120 ), etc. The Romans used a fraction system based on 12, including the Uncia which became the English word Ounce . Pre-decimalisation, Great Britain used a duodecimal currency system (12 Pence = 1 Shilling , 20 shillings or 240 pence to the Pound Sterling ), and Charlemagne established a monetary system that had a base of twelve and twenty, the remnants of which persist in many places. PLACES
FRACTIONS AND IRRATIONAL NUMBERS Duodecimal Fraction s are usually simple:
or complicated (A = ten, B = eleven)
As explained in Recurring Decimal s, whenever an Irreducible Fraction is written in “decimal” notation, in any base, the fraction can be expressed exactly (terminates) if and only if all the Prime Factor s 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: ¹⁄8 = ¹⁄(2×2×2), ¹⁄20 = ¹⁄(2×2×5), and ¹⁄500 (22×53) can be expressed exactly as 0.125, 0.05, and 0.002 respectively. ¹⁄3 and ¹⁄7, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, ¹⁄8 is exact; ¹⁄20 and ¹⁄500 recur because they include 5 as a factor; ¹⁄3 is exact; and ¹⁄7 recurs, just as it does in decimal. Arguably, factors of 3 are more commonly encountered in real-life Division problems than factors of 5 (or would be, were it not for the decimal system having influenced most cultures). Thus, in practical applications, the nuisance of Recurring 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 s, 11 (eleven) and 13 (thirteen), whereas ten is adjacent to 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, while 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, while only once in the factorization of ten; which means that most fractions whose denominators are Powers Of Two will have a shorter, more convenient terminating representation in dozenal than in decimal (e.g., 1/(22) = 0.25 dec = 0.3 doz; 1/(23) = 0.125 dec = 0.16 doz; 1/(24) = 0.0625 dec = 0.09 doz; 1/(25) = 0.03125 dec = 0.046 doz; etc.). As for Irrational Number s, none of them has a finite representation in ''any'' of the Rational -based positional number systems (such as the decimal and duodecimal ones); this is because a rational-based positional number system is essentially nothing but a way of expressing quantities as a sum of fractions whose denominators are powers of the base, and by definition no ''finite'' sum of rational numbers can ever result in an irrational number. For example, 123.456 = 1 × 103/10 + 2 × 102/10 + 3 × 10/10 + 4 × 1/10 + 5 × 1/102 + 6 × 1/103 (this is also the reason why fractions that contain prime factors in their denominator not in common with those of the base do not have a terminating representation in that base). Moreover, the infinite series of digits of an irrational number doesn't exhibit a pattern of recursion; instead, the different digits succeed in a seemingly random fashion. The following chart compares the first few digits of the decimal and duodecimal representation of several of the most important irrational numbers. As can be seen, it is easier to memorize the first nine digits of pi in base twelve than in base ten, while the opposite happens with the first ten digits of the number e: The first few digits of the decimal and dozenal representation of another important number, the Euler - Mascheroni Constant (the status of which as a rational or irrational number is not yet known), are: ADVOCACY AND "DOZENALISM" The case for the duodecimal system was put forth at length in F. Emerson Andrews' for its resemblance
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" because the latter comes from Latin roots that express twelve in base-ten terminology. EXTERNAL LINKS
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