Duodecimal system
The duodecimal system (also twelve system ) is a place value system for the representation of numbers . It uses the base twelve , so it is the "12-adic place value system". That means: Unlike the usual decimal system (with the base ten) there are twelve digits , so that a second digit is only required for natural numbers from twelve.
In the duodecimal system, the number 10 does not mean ten, but 1 dozen + 0 (i.e. twelve) and the number 0.1 does not mean a tenth, but a twelfth.
properties
No number less than twelve has such a good divisibility . Twelve has four non-trivial factors, 2, 3, 4 and 6, and it is a highly composite number . This has practical advantages when used as a sizing unit. In contrast, ten has only two non-trivial factors, 2 and 5.
The five most elementary fractions ( 1 ⁄ 4 , 1 ⁄ 3 , 1 ⁄ 2 , 2 ⁄ 3 and 3 ⁄ 4 ) all have a short, finite representation in the duodecimal system:
- 1 ⁄ 4 = 0.3 (12)
- 1 ⁄ 3 = 0.4 (12)
- 1 ⁄ 2 = 0.6 (12)
- 2 ⁄ 3 = 0.8 (12)
- 3 ⁄ 4 = 0.9 (12)
In the decimal system, these fractions are represented as follows:
- 1 ⁄ 4 = 0.25
- 1 ⁄ 3 = 0.33333 ...
- 1 ⁄ 2 = 0.5
- 2 ⁄ 3 = 0.66666 ...
- 3 ⁄ 4 = 0.75
The duodecimal system was occasionally referred to as the "optimal number system".
use
There are only a few cultures of which a duodecimal system is known. The division and the grouping into 12 is culturally very widespread and is evident in the concept of the dozen, the bulk (12 dozen), in the 12 hours twice a day, 12 signs of the zodiac , 12 signs in Chinese astrology, and the division old units of measurement (e.g. inches and feet ). However, it is not yet an indication of a duodecimal system.
In Roman numerals , the fractions are based on the base 12. The Roman name for a twelfth is Uncia , a word that later became the " ounce " measure of weight .
The spoken numbers of the plateau languages in Nigeria represent real duodecimal systems. The Nepalese language Chepang and the language Mahl of the indigenous population of the atoll Minicoy also use a duodecimal system.
In German and the other Germanic languages , the words for eleven and twelve are formed differently from the following numerals. Although there is no duodecimal system, it is interpreted as a linguistic indication that the decimal system of the Indo-Europeans may have mixed with a previously authoritative duodecimal system when the numerals were formed .
Groups that want to promote the awareness and use of the duodecimal system in modern times include the Dozenal Society of America (founded in 1944) and the Dozenal Society of Great Britain (founded in 1959).
Duodecimal counting with phalanges
In the usual decimal system (10 system) you count with the ten fingers (2 times 5) of both hands. In some areas of the world, however, there was a counting with the help of the phalanges , which leads to the number twelve with one hand, even to the number 144 (156) with two hands.
To do this, use the thumb of the main counting hand to touch the phalanges of the fingers in sequence from the little finger to the index finger (4 fingers x 3 phalanges each). With the other hand, the full dozen are then held in the same system.
See in detail one- and two-handed counting with phalanges and fingers .
The dual-decimal counting system on one hand is attested in India , Indochina , Pakistan , Afghanistan , Iran , Turkey , Iraq and Egypt .
Representation of numbers
Digits
In the duodecimal system, two more digits are required than in the decimal system. The Dozenal Society of Great Britain uses the characters suggested by Isaac Pitman in addition to the digits 0 to 92for ten and3for eleven (the digits 2 and 3 rotated 180 degrees).
The Dozenal Society of America uses for ten and for eleven instead . Where these characters are not available, X and E can be used as an alternative. The number with decimal representation 278 is thus written duodecimally as "1E2" .
Representation on computer systems
The characters and are contained in Unicode since version 8.0.0 (June 2015) as ↊ U + 218A turned digit two and ↋ U + 218B turned digit three in the block numerals based on a suggestion from 2013 as special characters without an intrinsic numeric value.
These characters can be represented in LaTeX by loading the package \usepackage{tipx}
as \textturntwo
or \textturnthree
.
These characters are also used in this article.
The characters and , however, are not available in any generally available character standard (as of June 2015). An application for inclusion in Unicode was not accepted in June 2013 regarding these characters. As a makeshift, they can be represented by the distantly similar characters (U + 1D4B3 mathematical script capital x ) and ℰ (U + 2130 script capital e ). (The Greek chi "χ" is less suitable, as it is a lowercase letter with descender and does not stand flush with other numeric characters.)
For the sake of simplicity, many computer programs for converting into different bases use the letters A and B for ten and eleven, based on the use in the hexadecimal system .
Whole and rational numbers
The representation of the numbers is similar to the representation in the commonly used decimal system , with the difference that the value of the digits is not determined by the corresponding power of ten, but by the appropriate power of twelve. For example, the sequence of digits 234 does not represent two hundred and thirty-four (as in the decimal system), but three hundred and twenty-eight, because in the duodecimal system the value is calculated as follows:
The indices indicate the basis used.
Duodecimal fractions are finite as in the decimal system, as is
- 1/2 = 0.6 (12)
- 1/3 = 0.4 (12)
- 1/4 = 0.3 (12)
- 1/6 = 0.2 (12)
- 1/8 = 0.16 (12)
- 1/9 = 0.14 (12)
- 1/12 = 0.1 (12)
or periodically, like
- 1/5 = 0.2497 (12)
- 1/7 = 0.186↊35 (12)
- 1/10 = 0.1 2497 (12)
- 1/11 = 0, 1 (12)
As in the decimal system, negative numbers are written with a preceding minus sign.
Basic arithmetic
Analogous to the numbers in the decimal system, the common arithmetic basic operations of addition , subtraction , multiplication and division can be carried out with duodecimal numbers . The algorithms required are basically the same, only the larger number of digits increases the multiplication table and the addition table.
* | 1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th | 9 | ↊ | ↋ | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th | 9 | ↊ | ↋ | 10 |
2 | 2 | 4th | 6th | 8th | ↊ | 10 | 12 | 14th | 16 | 18th | 1↊ | 20th |
3 | 3 | 6th | 9 | 10 | 13 | 16 | 19th | 20th | 23 | 26th | 29 | 30th |
4th | 4th | 8th | 10 | 14th | 18th | 20th | 24 | 28 | 30th | 34 | 38 | 40 |
5 | 5 | ↊ | 13 | 18th | 21st | 26th | 2↋ | 34 | 39 | 42 | 47 | 50 |
6th | 6th | 10 | 16 | 20th | 26th | 30th | 36 | 40 | 46 | 50 | 56 | 60 |
7th | 7th | 12 | 19th | 24 | 2↋ | 36 | 41 | 48 | 53 | 5↊ | 65 | 70 |
8th | 8th | 14th | 20th | 28 | 34 | 40 | 48 | 54 | 60 | 68 | 74 | 80 |
9 | 9 | 16 | 23 | 30th | 39 | 46 | 53 | 60 | 69 | 76 | 83 | 90 |
↊ | ↊ | 18th | 26th | 34 | 42 | 50 | 5↊ | 68 | 76 | 84 | 92 | ↊0 |
↋ | ↋ | 1↊ | 29 | 38 | 47 | 56 | 65 | 74 | 83 | 92 | ↊1 | ↋0 |
10 | 10 | 20th | 30th | 40 | 50 | 60 | 70 | 80 | 90 | ↊0 | ↋0 | 100 |
Convert to other place value systems
The first natural numbers are represented in the duodecimal system like this:
Duodecimal system | 0 | 1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th | 9 | ↊ | ↋ | 10 | 11 | 12 | 13 | 14th | 15th | 16 | 17th | 18th | 19th | 1↊ | 1↋ | 20th |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Decimal system | 0 | 1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th | 9 | 10 | 11 | 12 | 13 | 14th | 15th | 16 | 17th | 18th | 19th | 20th | 21st | 22nd | 23 | 24 |
From the duodecimal system to the decimal system
To get a decimal number from a duodecimal number, add up the specified multiples of the powers of 12, i.e. calculate the value of the number as specified in the definition of the 12-adic place value system:
- 234 (12) = 2 x 12 2 + 3 x 12 1 + 4 x 12 0 = 288 + 36 + 4 = 328.
From the decimal system to the duodecimal system
One way to convert a decimal number to the duodecimal system is to look at the division remainders that arise when the number is divided by the base 12.
In the example of 328 (10) it would look like this:
328: 12 = 27 Rest 4, 27: 12 = 2 Rest 3, 2: 12 = 0 Rest 2.
The digit sequence you are looking for can now be read from the remainder from bottom to top : 234 (12) .
Web links
Individual evidence
- ↑ George Douros: Unicode Fonts for Ancient Scripts. Retrieved June 19, 2015 .
- ↑ George Dvorsky: Why We Should Switch To A Base-12 Counting System . January 18, 2013. Accessed December 21, 2013.
- ^ Gerhardt, Ludwig (1987): Some remarks on the numerical systems of Plateau languages. In: Africa and Overseas 70: 19–29.
- ↑ Georges Ifrah: Universal History of Numbers . Licensed edition two thousand and one edition. Campus, Frankfurt am Main 1993, ISBN 978-3-86150-704-8 , Das Sexagesimalsystem, p. 69-75 u. 90–92 (French: Histoire universelle des chiffres . Translated by Alexander von Platen).
- ↑ Isaac Pitman (Ed.): A triple (twelve gross) Gems of Wisdom. London 1860.
- ^ A b Karl Pentzlin: Proposal to encode Duodecimal Digit Forms in the UCS. (PDF; 276 kB) ISO / IEC JTC1 / SC2 / WG2, Document N4399, March 30, 2013, accessed on June 29, 2013 (English).
- ↑ Scott Pakin: The Comprehensive LaTeX Symbol List. (PDF, 8.7 MB) January 19, 2017, p. 17 , archived from the original on September 28, 2017 ; Retrieved on September 28, 2017 (English, linking the original results in a mirror of CTAN , the archive link compare file: Comprehensive LaTeX Symbol list.pdf ).