Duodecimal

The duodecimal system (also known as base 12, dozenal, or rarely uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as '12' in the base ten numerical system) is instead written as '10' in duodecimal (meaning '1 dozen and 0 units', instead of '1 ten and 0 units'), whereas the digit string '12' means '1 dozen and 2 units' (i.e. the same number that in decimal is written as '14'). Similarly, in duodecimal '100' means '1 gross', '1000' means '1 great gross', and '0.1' means '1 twelfth' (instead of their decimal meanings '1 hundred', '1 thousand', and '1 tenth').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. The duodecimal system (also known as base 12, dozenal, or rarely uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as '12' in the base ten numerical system) is instead written as '10' in duodecimal (meaning '1 dozen and 0 units', instead of '1 ten and 0 units'), whereas the digit string '12' means '1 dozen and 2 units' (i.e. the same number that in decimal is written as '14'). Similarly, in duodecimal '100' means '1 gross', '1000' means '1 great gross', and '0.1' means '1 twelfth' (instead of their decimal meanings '1 hundred', '1 thousand', and '1 tenth'). The number twelve, a superior highly composite number, is the smallest number with four non-trivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range, and the smallest abundant number. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, and not 3, 4, or 6), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system. Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers (such as 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, ...) have a terminating representation in duodecimal. In particular, the five most elementary fractions (​ 1⁄2, ​ 1⁄3, ​ 2⁄3, ​ 1⁄4 and ​ 3⁄4) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (because it is the least common multiple of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary, octal and hexadecimal systems. Although the trigesimal and sexagesimal systems (where the reciprocals of all 5-smooth numbers terminate) do even better in this respect, this is at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize. Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Gure-Kahugu), Piti, and the Nimbia dialect of Gwandara; the Chepang language of Nepal and the Maldivian language (Dhivehi) of the people of the Maldives and Minicoy Island in India are known to use duodecimal numerals. Germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal. Historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, and the Babylonians had twelve hours in a day (although at some point this was changed to 24). Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. 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, and many other items 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. The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (three on each of four fingers). It is possible to count to 12 with the thumb acting as a pointer, touching each finger bone in turn. A traditional finger counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full. In a duodecimal place system twelve is written as 10, but there are numerous proposals for how to write ten and eleven. To allow entry on typewriters, letters such as A and B (as in hexadecimal), T and E (initials of Ten and Eleven), X and E (X from the Roman numeral for ten), or X and Z are used. Some employ Greek letters such as δ (standing for Greek δέκα 'ten') and ε (for Greek ένδεκα 'eleven'), or τ and ε. Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his book New Numbers an X and ℰ (script E, U+2130). Edna Kramer in her 1951 book The Main Stream of Mathematics used a six-pointed asterisk (sextile) ⚹ and a hash (or octothorpe) #. The symbols were chosen because they are available in typewriters, they also are on push-button telephones. This notation was used in publications of the Dozenal Society of America (DSA) in the period 1974–2008.