Duodecimal

[1] In these respects, duodecimal is considered superior to decimal, which has only 2 and 5 as factors, and other proposed bases like octal or hexadecimal.

In this system, one hand counts repeatedly to 12, while the other displays the number of iterations, until five dozens, i.e. the 60, are full.

[8] In the British Isles, this style of counting survived well into the Middle Ages as the long hundred.

Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches or 24 (12×2) Solar terms.

There are 12 inches in an imperial foot, 12 troy ounces in a troy pound, 12 old British pence in a shilling, 24 (12×2) hours in a day; many other items are counted by the dozen, gross (144, square of 12), or great gross (1728, cube of 12).

The Romans used a fraction system based on 12, including the uncia, which became both the English words ounce and inch.

Pre-decimalisation, Ireland and the United Kingdom used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the pound sterling or Irish pound), and Charlemagne established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.

Latin letters such as ⟨A, B⟩ (as in hexadecimal) or ⟨T, E⟩ (initials of Ten and Eleven) are convenient because they are widely accessible, and for instance can be typed on typewriters.

[11] Edna Kramer in her 1951 book The Main Stream of Mathematics used a ⟨*, #⟩ (sextile or six-pointed asterisk, hash or octothorpe).

[9][15] In March 2013, a proposal was submitted to include the digit forms for ten and eleven propagated by the Dozenal Societies in the Unicode Standard.

The most common method used in mainstream mathematics sources comparing various number bases uses a subscript "10" or "12", e.g. "5412 = 6410".

[25] The Dozenal Society of America suggested the pronunciation of ten and eleven as "dek" and "el".

In spite of the efficiency of these newer systems, terms for powers of twelve either already exist or remain easily reconstructed in English using words and affixes.

It should be made plain that the indices being a multiple of three, e.g. 10312 [1,00012], 10612 [1,000,00012], 10912 [1,000,000,00012] results, in these examples, in a great gross, a great-great-gross, and a great-great-great-gross, respectively.

[14][26] As numbers get larger (or fractions smaller), the last two morphemes are successively replaced with tri-mo, quad-mo, penta-mo, and so on.

[27][28] After hex-, further prefixes continue sept-, oct-, enn-, dec-, lev-, unnil-, unun-.

William James Sidis used 12 as the base for his constructed language Vendergood in 1906, noting it being the smallest number with four factors and its prevalence in commerce.

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.

However, the etymology of "dozenal" itself is also an expression based on decimal terminology since "dozen" is a direct derivation of the French word douzaine, which is a derivative of the French word for twelve, douze, descended from Latin duodecim.

Mathematician and mental calculator Alexander Craig Aitken was an outspoken advocate of duodecimal: The duodecimal tables are easy to master, easier than the decimal ones; and in elementary teaching they would be so much more interesting, since young children would find more fascinating things to do with twelve rods or blocks than with ten.

Anyone having these tables at command will do these calculations more than one-and-a-half times as fast in the duodecimal scale as in the decimal.

This is my experience; I am certain that even more so it would be the experience of others.But the final quantitative advantage, in my own experience, is this: in varied and extensive calculations of an ordinary and not unduly complicated kind, carried out over many years, I come to the conclusion that the efficiency of the decimal system might be rated at about 65 or less, if we assign 100 to the duodecimal.In "Little Twelvetoes," an episode of the American educational television series Schoolhouse Rock!, a farmer encounters an alien being with twelve fingers on each hand and twelve toes on each foot who uses duodecimal arithmetic.

The alien uses "dek" and "el" as names for ten and eleven, and Andrews' script-X and script-E for the digit symbols.

[32][33] Systems of measurement proposed by dozenalists include: The Dozenal Society of America argues that if a base is too small, significantly longer expansions are needed for numbers; if a base is too large, one must memorise a large multiplication table to perform arithmetic.

To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under positional notation).

Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values.

in the decimal 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.

, the nth power of the base b (although this includes the divisor 1, which does not produce fractions when used as the denominator).

[37] Thus, in practical applications, the nuisance of repeating decimals is encountered less often when duodecimal notation is used.

The following table gives the first digits for some important algebraic and transcendental numbers in both decimal and duodecimal.