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# Duodecimal numeral system

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The duodecimal numeral system, or dozenal[1] is a place-value notation using the powers of 12 rather than the powers of 10. In addition to using the digits 0 to 9, two additional digits are needed: the usual choice is the Latin letters A and B. Duodecimal, like decimal, owes its original usage to a fact of human anatomy, specifically, that the fingers of a hand (excluding the thumb) have twelve phalanges, "so that by using the thumb to count off these phalanges in turn, a person could count from 1 to 12."[2]

The number 201 in decimal, for example, is 149 in duodecimal, since ${\displaystyle 1\times 12^{2}+4\times 12^{1}+9\times 12^{0}=201}$. Among the first primes, in gross, i.e. base 12, representation, the digit B occurs last just thrice less often than either the digit 5 or the digit 7, but four times more often than the digit 1. (The digit 3 occurs this way of course just once, while 9 obviously never). In duodecimal, 1729 is 1001. The natural logarithm base ${\displaystyle \scriptstyle e\,}$ in duodecimal is 2.8752360698219BA71971... (see A027606).

Some have remarked "that from a mathematical point of view, the duodecimal system ... [has] several advantages over the decimal system in that the number 12 is divisible by 2, 3, 4, and 6, while 10 is divisible only by 2 and 5."[3] Nevertheless, it being easier to count ten fingers (including thumbs) on both hands than phalanges on one hand, the decimal system gained wider acceptance. Some aficionados of the Esperanto language have advocated changing over to duodecimal. But duodecimal will not replace decimal as the numeral system of choice for the same reason Esperanto will not replace English as the international language of choice.[4]

Nevertheless, there are many traces of duodecimal in the English language, such as the words dozen, gross and mass,[5] as well as in the British monetary system and in the system of American jurisprudence.

But perhaps the current decimal system is the perfect compromise between pragmatists who advocate bases of highly composite numbers and theoreticians who advocate prime numbers (like 7 or 11) as the basis of numeration as it would eliminate much ambiguity in the representation of fractions.[6]

## Notes

1. Weisstein, Eric W. "Duodecimal." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Duodecimal.html
2. Sergei Vasilievich Fomin, Number Systems Chicago: University of Chicago Press (1974) p. 4
3. Fomin, ibid., p. 4
4. "An esperantist is almost exactly equivalent to a duodecimalist: both are advocating changes that are logical, would be beneficial, and are not going to occur." Underwood Dudley, Mathematical Cranks. Cambridge: Cambridge University Press (1992): p. 23
5. Fomin, ibid. p. 4
6. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from French by David Bellos et al. London: The Harvill Press (1998): 41