Greek numerals

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Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Milesian numerals, Alexandrian numerals, or alphabetic numerals. 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 (cardinal) numbers, Hindu-Arabic numerals are used.

The earliest system of numerals in Greek were the acrophonic Attic numerals, operating much like Roman numerals (which derived from this scheme), with the following formula: Ι = 1, Π = 5, Δ = 10, ΠΔ = 50, Η = 100, ΠΗ = 500, Χ = 1000, ΠΧ = 5000, Μ = 10000 and ΠΜ = 50000.

Starting in the 4th century BC, the Attic numerals were 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 stigma ϛ or, in modern Greek, στ) for 6, qoppa (ϟ) for 90, and sampi (ϡ) for 900.^[1]. To distinguish numerals from letters they are followed by the "keraia" (Greek κεραῖα - antenna), a symbol similar to an acute sign (Unicode U+0734).

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 σμαʹ (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 "left karaia" (Unicode U+0375) is put in front of thousands to distinguish them from the standard use. For example, 2006 is represented as ͵βϛʹ (2000 + 6).

Letter	Value	Letter	Value	Letter	Value
αʹ	1	ιʹ	10	ρʹ	100
βʹ	2	κʹ	20	σʹ	200
γʹ	3	λʹ	30	τʹ	300
δʹ	4	μʹ	40	υʹ	400
εʹ	5	νʹ	50	φʹ	500
ϝʹ or ϛʹ or στʹ	6	ξʹ	60	χʹ	600
ζʹ	7	οʹ	70	ψʹ	700
ηʹ	8	πʹ	80	ωʹ	800
θʹ	9	ϟʹ	90	ϡʹ	900

In modern Greek, uppercase letters are preferred, as in Φίλιππος Βʹ = Philip II.

[edit] Higher numbers

The Greeks also used the myriad to denote 10,000 (Μʹ) and the myriad myriad for one hundred million (ΜΜʹ). In his text The Sand Reckoner the natural philosopher Archimedes proposed advanced ways to name very high numbers, such as the number of grains of sand on a beach, and the number of grains of sand on all the beaches on all the worlds in the universe.

[edit] Hellenistic zero

Hellenistic astronomers extended alphabetic Greek numerals 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".^[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.^[3]

[edit] References

1. ^ Numerals: Stigma, Koppa, Sampi
2. ^ Otto Neugebauer, The Exact Sciences in Antiquity (second edition, Providence, RI: Brown University Press, 1957) 13-14, plate 2.
3. ^ Ptolemy's Almagest, translated by G. J. Toomer, Book VI, (Princeton, NJ: Princeton University Press, 1998), pp.306-7)