Duodecimal

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Image:Duodecimal Multiplication Table.PNG

The duodecimal (also known as base-twelve or dozenal) system is a numeral system using twelve as its base.

The number 12 has four factors, which are 2, 3, 4 and 6. It is a more convenient number system for computing fractions compared with the decimal or vigesimal system.

The decimal system has only two factors, which are 2 and 5. Also the vigesimal system has four factors, which are 2, 4, 5 and 10; compared with the factor 3 (duodecimal) and 5 (vigesimal).

1 Origin
2 Places
- 2.1 Fractions
3 Advocacy and "dozenalism"
4 External links

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 Nigerian Middle Belt such as Janji, Kahugu, the Nimbia dialect of Gwandara, and the Chepang language of Nepal are known to use duodecimal numerals. In fiction, J. R. R. Tolkien's Elvish languages used the duodecimal.

In another hypothesis, twelve is the sum of ten fingers on hands and two feet.

Historically, the duodecimal is used in many civilizations, as units of time. 12 times 30 days are 360 days, i.e., one year. This is a standard usage of time, 30 days in a month, 12 months in a year. A clock has 12 hours, 12 means 360 days. Also 1 means 30 days, i.e., 1 month.

There are 12 traditional periods in a day in China and 12 signs of the zodiac in the horoscope. The Chinese calendar uses the Twelve Earthly Branches. It is believed that the observation of 12 appearances of the Moon in a year is the reason of this number.

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, are derived 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.

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, 50 means sixty (= five times twelve), 100 means one hundred forty-four (= twelve times twelve).

                                   Decimal Equivalent
       10  twelve (or a dozen)                   12
      100  one gross               12^2 =       144
     1000  one great gross         12^3 =     1 728
   10 000  twelve great gross      12^4 =    20 736
  100 000  ?                       12^5 =   248 832
1 000 000  ?                       12^6 = 2 985 984
      
       26  two and a half times twelve (= thirty)
       3B  three twelves and eleven (= forty-seven)
      1A6  square of twelve and ten twelves and six (= two hundred seventy)
      260  two and a half times gross (= three hundred sixty, one year)
      500  five gross (= 720 decimal, two years)
      700  seven gross (= 1008 decimal)
      B29  eleven gross two twelves and nine (= 1617 decimal)
     11B1  one great gross one gross eleven twelves and one (= 2005 decimal)
   36 A17  three dozen and six great gross ten gross one twelve and seven (= 74035 decimal)

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).

Fractions

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 a 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 (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 numbers 11(eleven) and 13(thirteen), whereas ten is adjacent to composite number 9.

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, Image:Scriptx.png and Image: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 Image:Scriptx.png for its resemblance to the Roman numeral X, and Image:Scripte.png as the first letter of the word "eleven".

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.