Греческие Числа: История и Развитие

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Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those in which Roman numerals are still used in the Western world. For ordinary cardinal numbers, however, modern Greece uses Arabic numerals.

The Minoan and Mycenaean civilizations' Linear A and Linear B alphabets used a different system, called Aegean numerals, which included number only symbols for powers of ten: 𐄇 = 1, 𐄐 = 10, 𐄙 = 100, 𐄢 = 1000, and 𐄫 = 10000. Attic numerals composed another system that came into use perhaps in the 7th century BCE. They were acrophonic, derived (after the initial one) from the first letters of the names of the numbers represented. They ran  = 1,  = 5,  = 10,  = 100,  = 1,000, and  = 10,000. The numbers 50, 500, 5,000, and 50,000 were represented by the letter with minuscule powers of ten written in the top right corner: , , , and
The present system probably developed around Miletus in Ionia. 19th century classicists placed its development in the 3rd century BCE, the occasion of its first widespread use. More thorough modern archaeology has caused the date to be pushed back at least to the 5th century BCE, a little before Athens abandoned its pre Eucleidean alphabet in favour of Miletus's in 402 BCE, and it may predate that by a century or two. The present system uses the 24 letters adopted under Eucleides, as well as three Phoenician and Ionic ones that had not been dropped from the Athenian alphabet (although kept for numbers): digamma, koppa, and sampi. The position of those characters within the numbering system imply that the first two were still in use (or at least remembered as letters) while the third was not. The exact dating, particularly for sampi, is problematic since its uncommon value means the first attested representative near Miletus does not appear until the 2nd century BCE, and its use is unattested in Athens until the 2nd century CE. (In general, Athenians resisted using the new numerals for the longest of any Greek state, but had fully adopted them by c. 50 CE. (That this was not the traditional location of sampi in the Ionic alphabetical order has led classicists to conclude that sampi had fallen into disuse as a letter by the time the system was created.) This alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example, 241 was represented as  (200 + 40 + 1). (It was not always the case that the numbers ran from highest to lowest: a 4th century BC inscription at Athens placed the units to the left of the tens. This practice continued in Asia Minor well into the Roman period.) In ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars: , , , etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as  (600 + 60 + 6). (Numbers larger than 1,000 reused the same letters but included various marks to note the change.) Fractions were indicated as the denominator followed by a keraia (ʹ); γʹ indicated one third, δʹ one fourth and so on. As an exception, special symbol ∠ʹ indicated one half, and γ°ʹ or γoʹ was two thirds. These fractions were additive (also known as Egyptian fractions); for example δʹ ϛʹ indicated
Although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the former ones, especially in the case of the obscure numerals. The old Q shaped koppa (Ϙ) began to be broken up ( and ) and simplified ( and ). The numeral for 6 changed several times. During antiquity, the original letter form of digamma (Ϝ) came to be avoided in favour of a special numerical one By the Byzantine era, the letter was known as episemon and written as or This eventually merged with the sigma tau ligature stigma ϛ ( or ). In modern Greek, a number of other changes have been made. Instead of extending an over bar over an entire number, the keraia (κεραία, lit. "hornlike projection") is marked to its upper right, a development of the short marks formerly used for single numbers and fractions. The modern keraia is a symbol similar to the acute accent (´), the tonos (U+0384,΄) and the prime symbol (U+02B9, ʹ), but has its own Unicode character as U+0374. Alexander the Great's father Philip II of Macedon is thus known as Φίλιππος Βʹ in modern Greek. A lower left keraia (Unicode: U+0375, "Greek Lower Numeral Sign") is now standard for distinguishing thousands: 2019 is represented as ͵ΒΙΘʹ (2 × 1,000 + 10 + 9). The declining use of ligatures in the 20th century also means that stigma is frequently written as the separate letters ΣΤʹ, although a single keraia is used for the group.

The practice of adding up the number values of Greek letters of words, names and phrases, thus connecting the meaning of words, names and phrases with others with equivalent numeric sums, is called isopsephy. Similar practices for the Hebrew and English are called gematria and English Qaballa, respectively.

In his text The Sand Reckoner, the natural philosopher Archimedes gives an upper bound of the number of grains of sand required to fill the entire universe, using a contemporary estimation of its size. This would defy the then held notion that it is impossible to name a number greater than that of the sand on a beach or on the entire world. In order to do that, he had to devise a new numeral scheme with much greater range. Pappus of Alexandria reports that Apollonius of Perga developed a simpler system based on powers of the myriad; was 10,000, was 10,0002 = 100,000,000, was 10,0003 = 1012 and so on. Note that the letter ο was still used with its original numerical value of 70; however, there was no ambiguity, as 70 could not appear in the fractional part of a sexagesimal number, and zero was usually omitted when it was the integer. Some of Ptolemy's true zeros appeared in the first line of each of his eclipse tables, where they were a measure of the angular separation between the center of the Moon and either the center of the Sun (for solar eclipses) or the center of Earth's shadow (for lunar eclipses). All of these zeros took the form , where Ptolemy actually used three of the symbols described in the previous paragraph. The vertical bar (|) indicates that the integral part on the left was in a separate column labeled in the headings of his tables as digits (of five arc minutes each), whereas the fractional part was in the next column labeled minute of immersion, meaning sixtieths (and thirty six hundredths) of a digit.