|If characters are not displayed correctly in the following, this is due to the font, see Display of Greek Numbers in Unicode and Help with Display Problems .|
|(2020 as milesic number)|
|(2020 as an acrophone number)|
The acrophone principle started with the first letters of the numerical words , while the other two forms of representation were based on the order of the letters in the alphabet , to which either decadic numerical values according to the Milesian principle or numerical values derived directly from their position in the alphabet according to the thesis principle were assigned. The latter, however, was not used by mathematicians because it is limited to only 24 values.
The alphabetical number system is an idea of the Greeks . Other alphabets were only later adapted according to their model. The principle of the system itself, however, is already in use in the hieratic and demotic spelling of Egyptian numbers . The alphabetical number system was by far the most important in ancient Greece. It was the standard number system used by all Greek mathematicians from ancient times to modern times , that is, until the adoption of Indian numerals in modern European mathematics.
In Greek it is still widely used today to write ordinal numbers (for example for rulers' names, comparable to the use of Roman numbers in German), while for cardinal numbers in Greek today, if emphasis is not placed on an emphatically traditional spelling, the decimal Indo-Arabic numerals are common. Every modern Greek is still familiar with the old numbers, for example the school classes are counted according to the old system: a child attending the fifth grade is therefore in the epsilon class.
According to the Greek model, almost all alphabets, for example the Hebrew alphabet , the Arabic alphabet , the Cyrillic alphabet and most other alphabets were adapted as numeric alphabets. However, since the Roman numerals were the standard number system in Westrom until the end of the Middle Ages, the Latin alphabet was never adapted until the modern era . Instead, the Eastern Roman Empire also used the Greek alphabetical numbers during the Middle Ages.
The acrophonic numbers
Around the beginning of the 5th century BC The use of acrophonic numbers is attested. The first letters of the number word are used to write the corresponding number value. The result is a system very similar to the - almost simultaneously developed - Roman numerals .
The following was valid: Ι = 1, Π (from pente) = 5, Δ (from deka) = 10, Η (from hekaton) = 100, Χ (from chilioi) = 1000 and Μ (from myrioi) = 10 000.
In addition to five equal pi like pente, there are also characters for 50, 500, 5000 and 50000. 10, 100, 1000 and 10000 are written into the pi, which corresponds to a multiplication by five, which avoids stringing the same number five times .
The alphabetic numbers
The linear defective system based on the thesis principle
In research, the “thesis system” refers to a linear numerical alphabet based on a formulation by Artemidor , in which the numerical values of the letters are derived directly from the position (thesis) of the respective letter in the order of the alphabet without any other increasing principle or other arithmetic operations surrender. As an indication that such a thesis system based on the principle alpha to omega = 1 to 24 (i.e. without the three special numeric characters digamma, also stigma, qoppa and sampi) could also have played a role in ancient Greek culture, In particular, the tradition of Homer's Iliad and Odyssey was assessed, the 24 chants of which are numbered and quoted according to the letters of the alphabet, at least in the tradition of the Alexandrian grammarians, as well as counting letters on the frieze tablets of the Pergamon Altar and for values less than ten (insofar without a clear Indicator for a thesis system) on lottery boards or to designate the districts of Alexandria .
The development of the number system
Since the middle of the 3rd millennium BC The so-called hieratic numbers are attested. It was about the contractions of the older, analogue representation of the Egyptian hieroglyphic numbers . In its original form, for example, the number 397 was represented as three little circles (three hundred) plus nine arcs (ninety) and seven lines. Each individual digit was initially separate, but was later merged into one character, a number. These digits were also used in the Rhind papyrus , for example .
The demotic script simplified the digits again. Middle of the 4th century BC Chr. , The Greeks came up with the idea, the first three to replace the group consisting of nine digits hieratic demotic series of numbers by the letters of their own alphabet. Since then one speaks of the "alphabetical number system". It divides the alphabet into three groups of nine characters each to represent the ones, tens and hundreds.
In order to have the total number of 3 × 9 = 27 characters required for this, three old letters that do not appear in the classical Greek alphabet were used as a model for three new characters for the purpose of displaying numbers.
- Digamma - It corresponds to the Latin F. The minuscule variant of the sigmoid (ς) is used together with a tau (τ) as a ligature (ϛ), which is also interpreted as a stigma . In today's printing works the letters sigma is usually tau (στ) used. 6 =
- Koppa - This is the old Qoph, i.e. the Latin Q. Originally written in the form ϙ, later also in the form: ϟ. 90 =
- 900 = Sampi - Sampi or Tsampi corresponds to the Phoenician Sade (San) and the Hebrew Tzade ; graphically in Greek: ϡ.
While F and Q take their original place in the alphabet, the old San or Tzade , which actually stands between P and Q, has been put in last place as Tsampi.
With these 27 characters and the numerical values assigned to them, the numbers 1 to 999 could be written by adding together units, tens and hundreds, i.e. 8 = η (Eta), 88 = πη (Pi + Eta = 80 + 8), 318 = τιη (Tau + Iota + Eta = 300 + 10 + 8). There was no sign for the zero and was also not required for the purposes of the number spelling, by adding about 200 = σ (Sigma), 202 = σβ (Sigma + Beta = 200 +2), 220 = σκ (Sigma + Kappa = 200 + 20) wrote.
In order to distinguish the numbers in the typeface from words, the former were usually overwritten with a line in the manuscripts, for example = 310, while in the age of letterpress an apostrophe ʹ (Δεξιά Κεραία) before the first digit was naturalized in Unicode U + 0374. But if the identity is clear as a number, it is sometimes dispensed with.
The numbers between 1000 and 9999 can also be displayed: For this purpose, the first number letter was multiplied by a thousand by adding a diacritical mark. In handwriting, one usually uses a sign that is in the form of a small hook open to the left, in front of the number at the top left. In letterpress, the subscript apostrophe ͵ (Αριστερή Κεραία) has prevailed, in Unicode U + 0375.
α β γ δ ε ϛ ζ η θ 1 2 3 4th 5 6th 7th 8th 9 ι κ λ μ ν ξ ο π ϟ 10 20th 30th 40 50 60 70 80 90 ρ σ τ υ φ χ ψ ω ϡ 100 200 300 400 500 600 700 800 900 ͵α ͵β ͵γ ͵δ ͵ε ͵ϛ ͵ζ ͵η ͵θ 1000 2000 3000 4000 5000 6000 7000 8000 9000
Extension of the number range
The standard extension to a hundred million
To this day the use of the numeral million is uncommon in Greece . Instead, one speaks of "a hundred myriads " (εκατομμύριο, ekatommýrio ), one myriad corresponds to the ten thousand. The number 99 million 999 thousand 999 were pronounced in ancient times 9999, myriads 9999. This is traditionally the highest number in the Greek number system.
This number can easily and unambiguously be written ʹΘϡϟθ M ʹΘϡϟθ ; where "M" means the myriads (a possible M = 40,000 does not occur in the system).
Likewise, for example, Aristarchus of Samos writes 71 million 755 thousand 875 as 7175 myriads 5875: ͵ZPOE M ͵EΩOE .
While Diophantos of Alexandria for example 4372 myriads 8097, the symbol for the myriads even reduced to one point: ͵ΔTOB . ͵HϟZ .
In order to clearly mark the myriad numbers, it was preferred in ancient times to write the units of the myriads using the acrophone Μ symbol.
|The number two myriads seven hundred four, for example, was therefore usually represented as follows:||ʹΨδ|
Today the numerals δισεκατομμύριο (disekatommýrio) = 1,000,000,000, τρισεκατομμύριο (trisekatommýrio) = 1,000,000,000,000 and so on.
The power extension to 10 to the power of 36
Another, but very seldom used system for the representation of large numbers can be found in Apollonios von Perge , who, according to the testimony of Pappus of Alexandria , differentiated myriads of the first, second, third, etc. to the ninth order in increasing power by using the Μ with overwritten the characters α to θ = 1 to 9, which consequently were no longer evaluated as a multiplier, but as the exponent of a power. To represent the number 5,462,360,064,000,000, this resulted in a spelling like the following:
|10000 3 × 5462||+||10000 2 × 3600||+||10000 1 × 6400|
- Franz Dornseiff : The alphabet in mysticism and magic. 2nd Edition. Teubner, Leipzig et al. 1925 ( Stoicheia 7, ), (reprint: Reprint-Verlag, Leipzig 1994, ISBN 3-8262-0400-X ).
- Gottfried Friedlein: The numerals and the elementary arithmetic of the Greeks and Romans and of the Christian West from the 7th to 13th century. Deichert, Erlangen 1869 (unchanged reprint: Sendet Reprint Verlag Wohlwend, Vaduz 1997).
- Georges Ifrah: Histoire universelle des ciffres. Seghers, Paris 1981, ISBN 2-221-50205-1 (German: Universal history of numbers. Campus Verlag, Frankfurt am Main et al. 1986, ISBN 3-593-33666-9 ).
- Karl Menninger : number and number. A cultural history of the number. 2nd revised and expanded edition. Vandenhoeck & Ruprecht, Göttingen 1958.
- Athanasius Kircher first describes such a Latinized system in his Oedipi Aegyptiaci, 1635, Volume II, First Part, page 488, but still without J, U & W, on a 23-letter basis. Therefore K = 10, T = 100 and Z = 500. For a few years there has also been a proposed 27-letter system, the so-called AJR system
- School of Mathematical and Computational Sciences University of St Andrews
- Unicode Character Code Charts: Greek and Coptic (Engl.), PDF