# Greek numerals: Wikis

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The numerical signs ʹ and ͵ redirect here. For the accent, ´, see acute accent.

Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Ionian numerals, Milesian numerals (i.e., from Miletus in Ionia), 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, Arabic numerals are used.

## History

Originally, before the adoption of the Greek alphabet, Linear A and Linear B had used a different system with symbols for 1, 10, 100, 1000 and 10000 operating with the following formula: | = 1, – = 10, ◦ = 100, ¤ = 1000, ☼ = 10000.[1]

The earliest alphabet-related system of numerals used with the Greek letters was a set of the acrophonic Attic numerals, operating much like Roman numerals (which derived from this scheme), with: Ι = 1, Π = 5, Δ = 10, Η = 100, Χ = 1000, Μ = 10000; and with 50, 500, 5000, and 50000 represented by composites of Π and a tiny version of the applicable power of ten.[1] The acrophonic system was replaced by a new alphabetic system, sometimes called the Ionic numeral system, from the 4th century BC.

## Description

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.[2]. To distinguish numerals from letters they are followed by the keraia (Greek κεραία, "hornlike projection"), a symbol ( ʹ ) similar to an acute sign ( ´ ) but with its own Unicode character (U+0374).

This 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). A famous example is 666 (the number of the Beast), which is represented as χξϛʹ (600 + 60 + 6) in medieval manuscripts of the Book of Revelation.

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 keraia" (Unicode U+0375, ‘Greek Lower Numeral Sign’) is put in front of thousands to distinguish them from the standard use. For example, 2008 is represented as ͵βηʹ (2000 + 8).

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.

## Higher numbers

To represent greater numbers, the Greeks also used the myriad from the old Attic numeral system in their notation. Its value is 10,000; the number of myriads was written above its symbol (Mʹ). For example (keraias replaced for technical reasons):

$\stackrel{\text{,}\delta\phi\pi\beta}{\Mu}\psi\theta' = 4582\times 10,000+709 = 45,820,709.$

Other forms are also possible. When that didn't suffice the myriad myriad (one hundred million, symbol: ΜΜʹ) was used[3].

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; it is described here.

## 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 ουδέν meaning "nothing".[4][5]

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.[6]

## References

1. ^ a b Systèmes numéraux en Grèce ancienne: description et mise en perspective historique
2. ^ Numerals: Stigma, Koppa, Sampi
3. ^ A Manual of Greek Mathematics by Thomas Little Heath, p.18.
4. ^ Otto Neugebauer, The Exact Sciences in Antiquity (second edition, Providence, RI: Brown University Press, 1957) 13-14, plate 2.
5. ^ Raymond Mercier, Consideration of the Greek symbol 'zero'PDF (1.32 MiB) Numerous examples
6. ^ Ptolemy's Almagest, translated by G. J. Toomer, Book VI, (Princeton, NJ: Princeton University Press, 1998), pp.306-7.

# Wiktionary

Up to date as of January 15, 2010

## English

### Noun

Greek numerals (singular Greek numeral)

1. The words used to represent numbers in the Greek language.
Cardinal numeral – the neuter form is used in counting; when describing number some decline for gender.
Ordinal numeral – have feminine (-η) and neuter (-ο) forms.
2. A system of numerals represented by letters of the Greek alphabet. These are used for special purposes, where in English Roman numerals might be used. The numerals are formed by the addition of a mark to the upper or lower case letter: Γ' δ' θ' In addition to the usual Greek letters three additional symbols are used: stigma (ϛ), koppa (ϟ) and sampi (ϡ); stigma (Ϛ or ϛ) may also be represented by ΣΤ' or στ'.
 Cardinal Ordinal Greek numerals Masculine Feminine Neuter Masculine 0 - - μηδέν - - - 1 ένας μία, μια ένα πρώτος Α' α' 2 - - δύο δυο δεύτερος Β' β' 3 τρεις τρεις τρία τρίτος Γ' γ' 4 τέσσερις τέσσερις τέσσερα τέταρτος Δ' δ' 5 - - πέντε πέμπτος Ε' ε' 6 - - έξι έκτος Σ' ΣΤ' σ' στ' 7 - - εφτά / επτά έβδομος Ζ' ζ' 8 - - οχτώ / οκτώ όγδοος Η' η' 9 - - εννέα / εννιά ένατος Θ' θ' 10 - - δέκα δέκατος Ι' ι' 11 - - ένδεκα / ένtεκα ενδέκατος ΙΑ' ια' 12 - - δώδεκα δωδέκατος ΙΒ' ιβ' 13 δεκατρείς δεκατρείς δεκατρία δέκατος τρίτος ΙΓ' ιγ' 14 δεκατέσσερις δεκατέσσερις δεκατέσσερα δέκατος τέταρτος ΙΔ' ιδ' 15 - - δεκαπέντε δέκατος πέμπτος ΙΕ' ιε'

# Simple English

Numeral systems by culture
Hindu-Arabic numerals
Western Arabic
Eastern Arabic
Khmer
Indian family
Brahmi
Thai
East Asian numerals
Chinese
Suzhou
Counting rods
Japanese
Korean
Alphabetic numerals
Armenian
Cyrillic
Ge'ez
Hebrew
Greek (Ionian)
Āryabhaṭa

Other systems
Attic
Babylonian
Egyptian
Etruscan
Mayan
Roman
Urnfield
List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64
1, 3, 9, 12, 20, 24, 30, 36, 60, more…

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 of the same way that Roman numerals are in the West; for ordinary (cardinal) numbers, Arabic numerals are used.

At first, before it was used more, the Greek alphabet, Linear A and Linear B had used a different system with symbols for 1, 10, 100, 1000 and 10000 operating with the following formula: | = 1, – = 10, ◦ = 100, ¤ = 1000, ☼ = 10000.[1]

The earliest alphabet-related system of numerals used with the Greek letters was a set of 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.

The acrophonic system was replaced by a new alphabetic system, sometimes called the Ionic numeral system, from the 4th century BC. 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: fau ϝ,(also used are stigma ϛ or, in modern Greek, στ) for 6, qoppa ϟ for 90, and sampi ϡ for 900.[2]. To distinguish numerals from letters they are followed by the "keraia" (Greek κεραίαinsect antenna), a symbol similar to an acute sign (Unicode U+0374).

This 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 keraia" (Unicode U+0375, ‘Greek Lower Numeral Sign’) is put in front of thousands to distinguish them from the standard use. For example, 2008 is represented as ͵βηʹ (2000 + 8).

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
ϛ 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 used more, as in Φίλιππος Βʹ = Philip II.

## Higher numbers

The Greeks also used the myriad to denote 10,000 (Μʹ) and the myriad myriad for one hundred million (ΜΜʹ).

Decimal Symbol Greek numeral
1 Ι ἴος (ios)
5 Π πέντε (pente)
10 Δ δέκα (deka)
100 Η ἧκατόν (hekaton)
1000 Χ χίλιοι (khilioi)
10000 Μ μύριοι (myrioi)