# Duodecimal

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

Since 2, 3, 4, 6 are factors of 12, it is a more convenient number system for computing fractions compared to decimal system, which has only the factors 2 and 5.

Languages based on a duodecimal 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 a duodecimal system.

Historically, the duodecimal system was used in many civilizations. The Romans, although they counted in base ten, used a duodecimal system to represent fractions. The Chinese use the 12 Earthly Branches. 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 periods in a day in China, 12 signs of the zodiac in horoscope, etc. Also, since there are 12 phalanx bones in the fingers of the human hand, it is possible to count up to 12 objects on the four fingers of one hand, using the thumb as indicator.

Many European languages have special words for 11 and 12 (and sometimes into the teens), which are often misinterpreted as vestiges of a base 12 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 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
3B  three dozen and eleven
TEE  ten gross eleven dozen and eleven
11E1  one great gross one gross eleven dozen and one (= the year 2005)
36 T17  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". In a hypothetical duo-decimal system, the term per gross (¹⁄144) might replace per cent (¹⁄100).

### 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 (T = ten, E = eleven)

• 1/5 = 0.24972497... recurring (easily rounded to 0.25)
• 1/7 = 0.186T35186T35... recurring (easily rounded to 0.187)
• 1/T = 0.124972497... recurring (rounded to 0.125)
• 1/E = 0.11111... recurring (rounded to 0.11)
• 1/11 = 0.0E0E... recurring (rounded to 0.0E)

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 0.002 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 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. Advocates of duo-decimal 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 is between two prime numbers 11 and 13, whereas 10 is adjacent to composite number 9.

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 contrary to the used symbols 'A' for ten and 'B' for eleven (or 'T' and 'E' for ten and eleven) as used in hexadecimal notation, he suggested in his book and used a script X and a script E, Missing image
Scriptx.png
image:Scriptx.png

and Missing image
Scripte.png
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 Missing image
Scriptx.png
image:Scriptx.png

for its resemblance to the Roman numeral X, and Missing image
Scripte.png
image:Scripte.png

as the first letter of the word "eleven".

There are modern advocacy groups that promote the use of the duo-decimal system, and they sometimes use the word dozenal, rejecting the duo-decimal as a word obviously based on the decimal counting system.

The Dozenal Society of America and Dozenal Society of Great Britain promote this base 12 system, arguing that it is better than the decimal system mathematically and in many other ways, however their membership and influence remains negligible.

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