Duodecimal

Duodecimal, sometimes called dozenal, is a base 12 number system.

The Dozenal Society of America and Dozenal Society of Great Britain promote that a base 12 system is better than the decimal system mathematically and in many other ways. Since 2, 3, 4, 6 are factors of 12, it is a convenient number in doing fractions. Compared to factor 2 and 5 in the decimal system, duodecimal seems to be more versatile.

Use of the base 12 number system is not common, but at least one example of duodecimal numerals is in use in the Chepang language of Nepal. Historically, the Romans, although they counted in base ten, used a duodecimal system to represent fractions.

Historically, the number 12 was used in many civilizations. It is believed that the observation of 12 appearances of the Moon in a year is the reason this number is used universally regardless of culture. Example of such usage include 12 months in a year, 12 hours on a clock, 12 traditional time divisions in a Chinese day, 12 signs of the zodiac in horoscope, 12 animal signs in Chinese astrology, etc. In many European languages, such as English, French, and German, the use of special names for 11 and 12 rather than names based on the decimal representation (such as twoteen) can be attributed to this rudimentary base-12 mindset.

Being a versatile denominator in fraction 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, 12 gross in a great gross, etc.

                                   Decimal Equivalent
       10  twelve (or a dozen)                   12
      100  one gross               12^2 =       144
     1000  one great gross         12^3 =      1728
   10,000  twelve great gross      12^4 =    20,736
  100,000  ?                       12^5 =   248,832
1,000,000  ?                       12^6 = 2,985,984
      
       15  a dozen and five
       3E  three dozen and eleven
      XEE  ten gross eleven dozen and eleven
     11E0  one great gross one gross eleven dozen (= the year 2004)
   36,X17  three dozen and six great gross ten gross one dozen and seven

Note that in English we say "a gross of apples", and not "a gross apples". The term per gross (¹⁄₁₄₄) would replace per cent (¹⁄₁₀₀).

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 decimal-based weights and measure or by the adoption of the duodecimal number system. In his book, he suggested and used a script X and a script E, and , 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 for its resemblance to the Roman numeral X, and as the first letter of the word "eleven".

Fractions

Duodecimal fractions are usually either very 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 (X = ten, E = eleven)

¹⁄₅ = 0.24972497 recurring (easily rounded to 0.25)

¹⁄₇ = 0.186X35186X35 recurring

¹⁄_X = 0.124972497 recurring (rounded to 0.125)

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-10 (= 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 0.005 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 base 10.

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 our culture). Thus, in practical applications, the nuisance of recurring decimals is encountered less often when duodecimal notation is used. This is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.

Duodecimal

Fractions

See also