Duodecimal

The duodecimal system (also Twelver System) is a positional notation to represent numbers. It uses the base twelve, so the " 12 -adic place value system ." This means that, unlike the common decimal (base 10), there are 12 digits, so that only natural numbers from 12 a second digit is required.

• 6.1 From the duodecimal to decimal
• 6.2 from decimal to duodecimal

Use and history

The number 12 had an important role in many cultures. It is considered the number of completeness. One reason is probably the (approximately) 12 lunar months in a year. Other examples are the use of twice 12 hours per day, 12 zodiac signs, 12 signs in Chinese astrology, 12 stars ( not derived from the number of founding states ) on the flag of the European Union. In many European languages ​​have their own name for the number 11 (" eleven " ) and 12 ( " twelve " ) instead of the regular decimal system name ( such as " zweiundzehn " or " two- ten "). This indicates as well as the use of the term dozen to a wider use of the base 12 number systems. In all Germanic languages ​​, the 13 is the first composite number, but the numbers from 0 to 12, inclusive, the rule applies in the German tender and only to use from 13 digits, according to Duden now deprecated and no longer binding, even if they have often is followed.

( 2, 3, 4, 6, 1 12 ) In addition, the 12 has the property of being divisible by relatively many numbers divisible, it is a highly composite number. This has certain practical advantages when used as a sizing (eg in inches and feet ).

A small drawback compared to the hexadecimal system, which the duodecimal to notify the decimal and the octal system is that the square root of the base is not an integer.

The duodecimal system is still used in some contexts;

Approaches to complement the decimal system with two extra digits to generally expect the duodecimal system, on the other hand could not prevail.

Duodezimales Counting with phalanxes

In the familiar decimal system (10 system) are counted with the fingers 10 ( 2 x 5 ) of both hands. In some areas of the world existed but counting with the help of the phalanges, which leads to number twelve one-handed, two-handed even with number 60. (See detail A - and two-handed counting with finger joints and fingers)

The fact that the Germans have, but then the words all numbers up to twelve own name with

• Composite number names ( thirteen, fourteen ...) and
• Derived numeric names (twenty two, thirty- three, eighty eight, one hundred, one thousand, ...)

Are formed (as well as the old quantity terms Gros for 144 = 12 * 12 and measure for 1728 = 12 * 12 * 12), creates a presumption that this counting system was also used by previous speakers of German.

The Duodezimalzählsystem on one hand is attested in India, Indo-China, Pakistan, Afghanistan, Iran, Turkey, Iraq and Egypt.

Duodezimales number system in spoken languages

The number systems of most natural languages ​​function according to the decimal system. While there is in German and other languages ​​individual terms such as " dozen " so alone but is still not a duodecimal ago. However, the spoken numbers of Plateau languages ​​in Nigeria represent real Duodezimalsysteme dar. Otherwise, no further evidence known.

Representation of numbers

Digits

The Dozenal Society of America (founded in 1944) in addition to the digits 0 to 9 are used even for 10 and 11 where these characters are not available, can the alternative, X and E are written. The number with decimal value 278 is thus written duodecimal as " 1E2 ".

The Dozenal Society of Great Britain (founded in 1959) instead uses the characters and ( the 180 degree rotated digits 2 and 3).

In this article, we use the number # and E for ten and eleven.

Representation on computer systems

The signs and are (as of June 2013) available in any standard characters generally available. An application for inclusion in Unicode was not adopted in June 2013 as to this character. Provisionally they can be represented by the distance similar character (U 1 D4B3 mathematical script capital x) and ℰ (U 2130 script capital e). ( The Greek Chi " χ " is less suitable, because it is not a lowercase letter with a descender flush with other numeric characters.)

The characters and can be represented in LaTeX as \ textturntwo or \ text turn three. Their inclusion in Unicode was applied for in 2013 and decided ( as a special character without intrinsic numerical value ), so that 2016 can be calculated as U 218 A and U 218 turned digit two digit three B turned in the block cipher with the availability. Many computer programs for converting to different bases used for the sake of simplicity, the letters A and B for ten and eleven in reference to the use in the hexadecimal system.

Whole and rational numbers

The representation of numbers is similar to the representation in the commonly used decimal system, with the difference that the significance of the digits is not determined by the appropriate power of ten, but by the matching twelve potency. For example, the digit string 234 is not (as in decimal ) represents the Zweihundertvierunddreißig, but the Dreihundertachtundzwanzig, because in the duodecimal the value is calculated by:

The indices in this case have to go on the base used.

Duodecimal fractions are as in the decimal either finite, as

, or periodically, such as

Negative numbers as you write in the decimal system, preceded by a minus sign.

Basic arithmetic

Quite analogously to the numbers in the decimal system can be personalized with Duodezimalzahlen the usual basic arithmetic operations of addition, subtraction, multiplication and division to perform. The algorithms required are basically the same, only by the greater number of digits the basics and the addition table is greater.

Convert to other value systems

The first natural numbers are represented in the duodecimal as:

From the duodecimal to decimal

To get from a Duodezimalzahl a decimal number, you count along the specified multiple of the 12- potencies, ie calculates the value of the number as specified on the definition of the 12 -adic value system:

From decimal to duodecimal

A possibility of converting a decimal number in the duodecimal, is the consideration of the division radicals which are formed when the number is divided by the base 12.

In the example of the 328 (10 ) which would look like this:

328: 12 = 27 remainder 4,   27: 12 = 2 remainder 3,    2: 12 = 0 remainder 2 The desired sequence of digits you now read from bottom to top on the residues from: 234 (12).