Addition system
An addition system is a number system in which the value of a number is calculated by adding the values of its digits . In contrast, the position of the digits also plays a role in a place value system .
A simple example of an addition system is the tally sheet , a unary system : There is only one digit, for example the vertical bar "|". A number is represented as a sequence of bars, where the value of a number corresponds to the number of bars. For example, the decimal number 3 is used in this system as ||| written. However, such a notation quickly becomes confusing for large numbers, so that the need arises to introduce additional digits.
Various additive number fonts have been developed over the years . The Roman numerals still in use today have 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 subtraction notation, it plays The order of the digits in a Roman number does not matter to the value of the number, although it is common to order the digits in descending order from left to right. In principle, however, the three numbers XII, IXI, IIX are equivalent and correspond to the decimal 12 (2 * 1 + 10).
In addition systems, adding numbers is quite easy, since the digits of the summands are simply combined to form a new number. If necessary, groups of digits are then combined into more significant digits. There is no need to remember carry-overs, as is necessary in place value systems. 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 of digits is difficult. Because if W is the largest digit value, then you need at least Z / W digits to represent a very large number Z. The relationship between length and value is ( asymptotically ) linear - in contrast to the place value systems, in which it is logarithmic .
Developed addition systems
Different digits for each power of the base
Such a number system was already used about 5000 years ago in ancient Egypt with the hieroglyphic numbers. The principle of this system sets a number for each power of the base, e.g. E.g .: E = 1, Z = 10, H = 100 and T = 1000.
The individual positions were mostly arranged graphically; in the following, basic example according to the domino eyes.
HHH ZZZ E
1982 = T HHH Z Z
HHH ZZZ E
Almost at the same time - that is, during the Proto- Elamite epoch - such a number system was developed in Susa , just like - from the second millennium BC - by the Minoans on Crete, and a little later also by the Hittites . Number systems based on this principle are also known from Meso-American high cultures.
The disadvantage of this system is that each digit consists of the analog repetition of the same sign, which is why the ancient Egyptians merged each digit into a single digit by hieratic handwriting as early as the middle of the third millennium . These hieratic numbers served as a model for the later alphabetical numbers.
More than one digit within the same power of the base
Using your own characters for the “half numbers” prevents the same character from being repeated too often.
An example of this are the Roman numerals , which in addition to the letters I, X, C and M as symbols for 1, 10, 100 and 1000 also use V, L and D for 5, 50 and 500.
The digits are written and added with decreasing value. For example, 1776 is represented as MDCC.LXXVI. In order to keep the numbers a little shorter, the system was later modified so that each digit can appear a maximum of three times in a row. If there is a smaller number in front of a larger one, the former is subtracted from the latter. So VIIII became IX. This subtraction rule within the addition system is not always taken to heart.
In Western Europe, the Roman number system was widely used until the 15th century.
Separate digit for each multiplicity within a power of the base
The hieratic numbers (see above), like the decimal system, already obeyed the principle of different digits to identify each frequency of occurrence (multiplicity) of a (used) power of the base. However, each power of the base still had its own (nine) digits for its possible multiplicities, which differ from the digits for the other powers. Numbers for the frequency of unused powers of the base - i.e. one or more zeros - do not yet exist. There were a total of 36 (4 × 9) hieratic symbols for the numbers 1 to 9999.
In the middle of the fourth century BC, the ancient Greeks created the so-called alphabetic numbers based on these hieratic numbers by replacing the first 3 × 9 hieratic numbers with the letters of their alphabet. With the hybrid use of the acrophone numbers , large numbers can also be displayed.
Except in the western Roman areas, where the Roman numerals were always used, this progressive system - in its adaptations to the respective alphabets - dominated the science and administration of Persia , Armenia , Georgia , Arabia , Ethiopia , the Byzantine Empire and the ancient Russia . Only the Indian numerals gradually replaced the system after four thousand years of dominance. In the Arab region as early as the end of the first millennium, otherwise not until the middle of the second millennium.
Individual evidence
- ↑ Ancient Egyptian hieratic numbers ( Memento from March 2, 2016 in the Internet Archive ) (graphic)