Sistema de numeração duodecimal

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A duodecimal multiplication table

O sistema de numeração duodecimal (também conhecido como base-12) é um sistema de numeração que usa o número 12 como sua base.

O número 12 possui seis fatores, que são 1, 2, 3, 4, 6 e 12. É um sistema de numeração mais conveniente para a computação comparado com o sistema decimal e o vigesimal.

[editar] Origem

A origem do sistema é um conjunto de relações que tem como base a abertura de um compasso e seu círculo o sistema pode ser complementado com a base o sistema decimal 10 ou hexagonal 60. Com a mesma abertura de 60 graus uma pessoa da pré-história podia dividir uma circunferência em exatos doze ou seis partes. Por razões praticas ficou definido que esse tipo de divisão funcionasse para dividir o tempo uma hora ou um mes equivale a 1/12 de circunferência e é dividida em sessenta unidades chamadas minuto.

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 Nigerian 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:

• the approximate number of lunar months in an Earth year;
• the sum of ten fingers on human hands and two feet; or
• the number of phalanx bones in the four fingers of one hand, with the thumb used as an indicator.

Historically, units of time in many civilizations are duodecimal, which may come as a generalization of the use for months. There are twelve signs of the zodiac. There are twelve European hours in a day or night. Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches.

Many European languages have special words for 11 and 12 (and sometimes into the teens), which are often misinterpreted as vestiges of a base-twelve system. However, in actuality, most if not all of these terms have been eroded from decimal roots. For example, in Latin, the teens were formed by suffixing -decem (ten) to the respective words. In the modern Romance languages, this is often obscured by sound changes. For example, undecem and duodecem became, in Spanish, once and doce (likewise trece, catorce, quince). English “eleven” and “twelve” are believed to come from Proto-Germanic *ainlif and *twalif (respectively “one left” and “two left”), also related to base-ten. Admittedly, the survival of such apparently unique terms may be connected with duodecimal tendencies, but their origin is not duodecimal.

Being a versatile denominator in fractions may explain why we have 12 inches 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.

[editar] Places

In a duodecimal place system, ten is written as A, eleven is written as B, twelve is written as 10. According to this notation, duodecimal 50 expresses the same quantity as decimal 60 (= five times twelve), duodecimal 60 is equivalent to decimal 72 (= six times twelve = half a gross), duodecimal 100 has the same value as decimal 144 (= twelve times twelve = one gross), etc. Note that in English we say a gross of apples, and not *a gross apples. In a hypothetical duodecimal system, the term per gross (¹⁄₁₄₄) might replace per cent (¹⁄₁₀₀).

[editar] Fractions and irrational numbers

Duodecimal fractions are usually simple:

• 1/2 = 0.6
• 1/3 = 0.4
• 1/4 = 0.3
• 1/6 = 0.2
• 1/8 = 0.16
• 1/9 = 0.14

or complicated (A = ten, B = eleven)

• 1/5 = 0.24972497... recurring (easily rounded to 0.25)
• 1/7 = 0.186A35186A35... recurring (easily rounded to 0.187)
• 1/A = 0.124972497... recurring (rounded to 0.125)
• 1/B = 0.11111... recurring (rounded to 0.11)
• 1/11 = 0.0B0B... recurring (rounded to 0.0B)

As explained in recurring decimals, 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 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: ¹⁄₈ = ¹⁄_(2×2×2), ¹⁄₂₀ = ¹⁄_(2×2×5), and ¹⁄₅₀₀ = ¹⁄_{(2×2×5×5×5)} 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 numbers, 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 numbers, 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:

[editar] Advocacy and "dozenalism"

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. In contrary to the symbols 'A' for ten and 'B' for eleven as used in hexadecimal notation and vigesimal notation (or 'T' and 'E' for ten and eleven), he suggested in his book and used a script X and a script E, Imagem:Scriptx.png and Imagem:Scripte.png, to represent the digits ten and eleven respectively, because, at least on a page of Roman script, these characters were distinct from any existing letters or numerals, yet were readily available in printers' fonts. He chose Imagem:Scriptx.png for its resemblance to the Roman numeral X, and Imagem:Scripte.png as the first letter of the word "eleven". Another popular notation, introduced by Isaac Pitman, is to use an inverted 2 to represent ten and an inverted 3 to represent eleven; this is the system commonly employed by the Dozenal Society of Great Britain and has the advantage of being easily recognizable as digits. On the other hand, the Dozenal Society of America adopted for some years the convention of using an asterisk * for ten and a hash # for eleven (the reason was they are present in telephone dials); however, critics pointed out these symbols do not look anything like digits.

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.

[editar] External links

• Decimal vs. Duodecimal: An interaction between two systems of numeration — duodecimal numerals in languages in Nigerian Middle Belt
• The origin of a duodecimal system (Japanese) — explains a possible origin of a duodecimal system in a language
• Dozenal Society of America
• Dozenal Society of Great Britain website