Greek numerals

Template:Table Numeral Systems Greek numerals are a system of representing numbers using letters of the Greek alphabet. In modern Greece, they are still in use for ordinal numbers, and in much the same situations as Roman numerals are in the West; for ordinary numbers, Arabic numerals are used.

The earliest system of numerals in Greek was acrophonic, operating much like Roman numerals (which derived from this scheme), with the following acrophonic formula: Ι = 1, Π = 5, Δ = 10, Η = 100, Χ = 1000, and Μ = 10000.

Starting in the 4th century BC, the acrophonic system was replaced with a quasi-decimal alphabetic system, sometimes called the Ionic numeral system. Each unit (1, 2, ..., 9) was assigned a separate letter, each tens (10, 20, ..., 90) a separate letter, and each hundreds (100, 200, ..., 900) a separate letter. This requires 27 letters, so the 24-letter Greek alphabet was extended by using three obsolete letters: digamma (Ϝ, also used are ς or στ) for 6, qoppa (Ϟ) for 90, and sampi (Ϡ) for 900. An acute sign (´) is used to distinguish numerals from letters.

The alphabetic system operates on the additive principle in which the numeric values of the letters are added together to form the total. For example, 241 is represented as σμα´ (200 + 40 + 1).

To represent numbers from 1,000 to 999,999 the same letters are reused to serve as thousands, tens of thousands, and hundreds of thousands. A comma or inverted acute is put in front of thousands to distinguish them from the standard use. For example, 2004 is represented as ,βδ´ (2000 + 4).

Letter	Value	Letter	Value	Letter	Value
α´	1	ι´	10	ρ´	100
β´	2	κ´	20	σ´	200
γ´	3	λ´	30	τ´	300
δ´	4	μ´	40	υ´	400
ε´	5	ν´	50	φ´	500
ϝ´ or ς´ or στ´	6	ξ´	60	χ´	600
ζ´	7	ο´	70	ψ´	700
η´	8	π´	80	ω´	800
θ´	9	ϟ´	90	ϡ´	900

Hellenistic zero

Hellenistic astronomers extended this into a sexagesimal positional numbering system by limiting each position to a maximum value of 50 + 9 and including a special symbol for zero, which was also used alone like our modern zero, more than as a simple placeholder. However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number. This system was probably adapted from Babylonian numerals by Hipparchus c. 140 BC. It was then used by Ptolemy (c. 140), Theon (c. 380), and Theon's daughter Hypatia (died 415).

The Greek sexagesimal place holder or zero symbol changed over time. The symbol used on papyri during the second century was a very small circle with an overbar several diameters long, terminated or not at both ends in various ways. Later, the overbar shortened to only one diameter, similar to our modern o macron (ō), which was still being used in late medieval Arabic manuscripts whenever alphabetic numerals were used. But the overbar was omitted in Byzantine manuscripts, leaving a bare ο (omicron). This gradual change from an invented symbol to ο does not support the hypothesis that the latter was the initial of ουδεν meaning "nothing". [Otto Neugebauer, The Exact Sciences in Antiquity (second edition, Providence, RI: Brown University Press, 1957) 13-14, plate 2.]

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 0 | 0 0, 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 minutes of immersion, meaning sixtieths (and thirty-six-hundredths) of a digit. [Ptolemy's Almagest, translated by G. J. Toomer, Book VI, (Princeton, NJ: Princeton University Press, 1998), pp.306-7)]

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