Sign-value notation
An addition system is a number system in which calculates the value of a number by adding the values of its digits. In contrast, the position of the number plays a role in a priority system.
A simple example of an addition system is the tally sheet, a Unärsystem: There is only one digit, for example, the vertical line "| ". A number is represented as a sequence of strokes, where the value of a count of the number of strokes equal. The decimal number 3, for example, in this system as | | | written. Such a notation is but quickly become confusing, so the need arises in large numbers to introduce further digits.
In the Roman numerals, there are seven digits namely I, V, X, L, C, D, and M. These correspond to the values 1, 5, 10, 50, 100, 500 and 1000. With the exception of the sequence makes the Subtraktionsschreibweise digits of a Roman numeral no difference to the value of the number, although it is customary to arrange the digits from left to right in descending order. In principle, the three figures XII, IXI, IIX equivalent and correspond to the decimal 12 ( 2 * 1 10).
In addition, the systems adding numbers fall quite easily, since the number of summands need to be simply pulled together to form a new number. Then, if appropriate, summarizes groups of digits to higher-value digits. Memorizing of carries, as it is necessary in value systems is eliminated. The disadvantage of addition systems is that multiplication, fractions and generally higher mathematics are difficult to accomplish. In particular, the representation of very large numbers with a necessarily finite number stock is difficult. Because W is the largest numeric value, then you need to represent a very large number Z is at least Z / W digits. The relationship between length and value is therefore ( asymptotically ) linear - in contrast to the value systems in which he is logarithmic.
Developed addition systems
Various digits for each power of the base
Such a number system was used with the hieroglyphic figures in ancient Egypt about 5000 years ago already.
The principle of this system is for each power of the base a digit, eg: E = 1, Z = 10, H = 100 and T = 1000th The individual sites were mostly arranged graphically; in the following, basic example of the Domino eyes.
HHH ZZZ E 1982 = T HHH Z Z HHH ZZZ E Such a number system developed, as well as - - In Susa was almost the same time - so even during the proto- Elamite period from the second millennium BC - from the Minoans in Crete and a little later by the Hittites. From meso- American civilizations are number systems based on this principle also known.
The disadvantage of this system is that each point consists of the analog repetition of the same character, which is why the ancient Egyptians moved in together already in the middle of the third millennium any body hieratic - handwritten to a single digit. This hieratic figures were later alphabetical ratings of the model.
More than one digit in the same power of the base
The use of own characters for the " half- numbers " avoid too frequent repetition of the same character.
An example of this are the Roman numerals that use in addition to the letters I, X, C and M as symbols for 1, 10, 100 and 1000, V, L, and D 5, 50 and 500.
The numbers are written in descending order of importance and added. 1776 is shown as only MDCC.LXXVI. To keep the numbers a little shorter, the system was later modified so that each digit must occur consecutively more than three times. Is a smaller number in front of a larger, the former is withdrawn from the latter. Thus VIIII to IX. This subtraction within the Addition system is not always heeded.
In Western Europe, the Roman numeral system was widely used until the 15th century.
Each multiplicity within a power of the base has its own number
Already the hieratic figures (see above) as the decimal obeyed the principle of different numbers to identify each frequency of occurrence ( multiplicity) a ( used ) power of the base. Each had power of the base but still have their own (nine) digits for their possible multiplicities, which are different from the numbers for the other powers. Figures for the incidence of unused powers of the base - ie one or more zeros - however, do not yet exist. Thus there were a total of 36 (4 × 9) hieratic symbols for the numbers 1 to 9999. ( See also links. )
Middle of the fourth century before Christ created the ancient Greeks, starting from these hieratic figures, the so-called index numbers by replacing the first 3 × 9 hieratic figures by the letters of their alphabet. Using the hybrid use of akrophonen numbers, even large numbers are represented.
Except in the western Roman territories, where you always clung to the Roman numerals, dominated this progressive system - in their adaptations to the respective alphabets - a very long time the science and management of Persia, Armenia, Georgia, Arabia, Ethiopia, the Byzantine Empire and the old Russia. Only the Hindu numerals solved the system, after four thousand years of dominance, gradually. In the Arab world at the end of the first millennium, or until the middle of the second millennium.