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GREEK NUMERALS are a system of writing numbers using the letters of the Greek alphabet
Greek alphabet
. These alphabetic numerals are also known as IONIC or IONIAN NUMERALS, MILESIAN NUMERALS, or ALEXANDRIAN NUMERALS. In modern Greece
Greece
, they are still used for ordinal numbers and in contexts similar to those in which Roman numerals
Roman numerals
are still used elsewhere in the West. For ordinary cardinal numbers , however, Greece uses Arabic numerals
Arabic numerals
.

CONTENTS

* 1 History * 2 Description * 3 Table * 4 Higher numbers * 5 Zero * 6 See also * 7 References * 8 External links

HISTORY

The Minoan and Mycenaean civilizations ' Linear A
Linear A
and Linear B alphabets used a different system, called Aegean numerals, which included specialized symbols for numbers: 𐄇 = 1, 𐄐 = 10, 𐄙 = 100, 𐄢 = 1000, and 𐄫 = 10000.

Attic numerals, which were later adopted as the basis for Roman numerals , were the first alphabetic set. 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, = 1000, and = 10000. 50, 500, 5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner: , , , and . The same system was used outside of Attica
Attica
, but the symbols varied with the local alphabets : in Boeotia
Boeotia
, was 1000.

The present system probably developed around Miletus
Miletus
in Ionia. 19th-century classicists placed its development in the 3rd century BC, 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 BC, a little before Athens abandoned its pre-Euclidean alphabet in favor of Miletus
Miletus
's in 402 BC, and it may predate that by a century or two. The present system uses the 24 letters adopted by Euclid
Euclid
as well as three Phoenician and Ionic ones that were not carried over: 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
Miletus
does not appear until the 2nd century BC and its use is unattested in Athens until the 2nd century AD. (In general, Athens resisted the use of the new numerals for the longest of any of the Greek states but had fully adopted them by AD c. 50. )

DESCRIPTION

_ Greek numerals
Greek numerals
in a c. 1100 Byzantine manuscript of Hero of Alexandria
Alexandria
's Metrika_. The first line contains the number "͵θϡϟϛ δʹ ϛʹ", i.e. " 9996 4⁄6". It features each of the special numeral symbols sampi (ϡ), koppa (ϟ), and stigma (ϛ) in their minuscule forms. _ A 14th-century Byzantine map of the British Isles from a manuscript of Ptolemy
Ptolemy
's Geography_ , using Greek numerals for its graticule : 52–63°N of the equator and 6–33°E from Ptolemy's Prime Meridian
Prime Meridian
at the Fortunate Isles.

Greek numerals
Greek numerals
are decimal , based on powers of 10. The units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta . Instead of reusing these numbers to form multiples of the higher powers of ten, however, each multiple of ten from 10 to 90 was assigned its own separate letter from the next nine letters of the Ionic alphabetfrom iota to koppa . Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well, from rho to sampi . (The fact that this was _not_ the traditional location of sampi or its possible predecessor san has led classicists to conclude that it was no longer in use even locally 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
Asia Minor
well into the Roman period . ) In ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars : α, β, γ, etc. In medieval manuscripts of the Book of Revelation
Book of Revelation
, the number of the Beast 666 is written as χξϛ (600 + 60 + 6). (Numbers larger than 1,000 reused the same letters but included various marks to note the change.)

Although the Greek alphabet
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 ( ) came to be avoided in favor 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 ( or ).

In modern Greek , a number of other changes have been made. Instead of extending an overbar 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 (ʹ) similar the acute accent (´) but has its own Unicode
Unicode
character as U+0374. Alexander the Great
Alexander the Great
's father Philip II of Macedon
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: 2015 is represented as ͵ΒΙΕʹ (2000 + 10 + 5).

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 art of assigning Greek letters also being thought of as numerals and therefore giving words/names/phrases a numeric sum that has meaning through being connected to words/names/phrases of similar sum is called isopsephy (gematria ).

TABLE

ANCIENT BYZANTINE MODERN VALUE

ANCIENT BYZANTINE MODERN VALUE

ANCIENT BYZANTINE MODERN VALUE

ANCIENT BYZANTINE MODERN VALUE

α Αʹ 1

ι Ιʹ 10

ρ Ρʹ 100 ">β Βʹ 2

κ Κʹ 20

σ Σʹ 200

͵β ͵Β 2000

γ Γʹ 3

λ Λʹ 30

τ Τʹ 300

͵ ͵Γ 3000

δ Δʹ 4

μ Μʹ 40

υ Υʹ 400

͵ ͵Δ 4000

ε Εʹ 5

ν Νʹ 50

φ Φʹ 500

͵ε ͵Ε 5000

& ">ξ Ξʹ 60

χ Χʹ 600

͵ & ͵ ͵ ">ζ Ζʹ 7

ο Οʹ 70

ψ Ψʹ 700

͵ζ ͵Z 7000

η Ηʹ 8

π Πʹ 80

ω Ωʹ 800

͵η ͵H 8000

θ Θʹ 9

& & Ϟʹ 90

& & &

& "> M o {displaystyle {stackrel {rho oepsilon }{mathrm {M} }}} ͵εωοεʹ for 1,755,875.

HIGHER NUMBERS

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.

ZERO

Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus

Hellenistic astronomers extended alphabetic Greek numerals
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 today's 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
Babylonian numerals
by Hipparchusc. 140 BC. It was then used by Ptolemy
Ptolemy
(c. 140), Theon (c. 380) and Theon's daughter Hypatia
Hypatia
(murdered 415).

In Ptolemy\'s table of chords , the first fairly extensive trigonometric table, there were 360 rows, portions of which looked as follows: ' ` o {displaystyle {begin{array}{ccc}pi varepsilon varrho iota varphi varepsilon varrho varepsilon iota {tilde {omega }}nu &varepsilon {overset {text{'}}{nu }}vartheta varepsilon iota {tilde {omega }}nu &{overset {text{`}}{varepsilon }}xi eta kappa mathrm {o} sigma tau {tilde {omega }}nu \{begin{array}{l}hline pi delta angle '\pi varepsilon \pi varepsilon angle '\hline pi mathrm {stigma} \pi mathrm {stigma} angle '\pi zeta \hline end{array}}&{begin{array}{rrr}hline pi &mu alpha &gamma \pi alpha &delta &iota varepsilon \pi alpha &kappa zeta &kappa beta \hline pi alpha &nu &kappa delta \pi beta &iota gamma &iota vartheta \pi beta &lambda mathrm {stigma} &vartheta \hline end{array}}&{begin{array}{rrrr}hline circ &circ &mu mathrm {stigma} &kappa varepsilon \circ &circ &mu mathrm {stigma} &iota delta \circ &circ &mu mathrm {stigma} &gamma \hline circ &circ &mu varepsilon &nu beta \circ &circ &mu varepsilon &mu \circ &circ &mu varepsilon width:51.526ex; height:26.176ex;" alt="{begin{array}{ccc}pi varepsilon varrho iota varphi varepsilon varrho varepsilon iota {tilde {omega }}nu &varepsilon {overset {text{}}{nu }}vartheta varepsilon iota {tilde {omega }}nu &{overset {text{`}}{varepsilon }}xi eta kappa mathrm {o} sigma tau {tilde {omega }}nu \{begin{array}{l}hline pi delta angle \pi varepsilon \pi varepsilon angle \hline pi mathrm {stigma} \pi mathrm {stigma} angle \pi zeta \hline end{array}}&{begin{array}{rrr}hline pi &mu alpha &gamma \pi alpha &delta &iota varepsilon \pi alpha &kappa zeta &kappa beta \hline pi alpha &nu &kappa delta \pi beta &iota gamma &iota vartheta \pi beta &lambda mathrm {stigma} &vartheta \hline end{array}}&{begin{array}{rrrr}hline circ &circ &mu mathrm {stigma} &kappa varepsilon \circ &circ &mu mathrm {stigma} &iota delta \circ &circ &mu mathrm {stigma} &gamma \hline circ &circ &mu varepsilon &nu beta \circ &circ &mu varepsilon &mu \circ &circ &mu varepsilon however, there was no ambiguity, as 70 could not appear in the fractional part of a 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
Moon
and either the center of the Sun
Sun
(for solar eclipses ) or the center of Earth
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 _minute of immersion_, meaning sixtieths (and thirty-six-hundredths) of a digit.

SEE ALSO

* Attic numerals * Gematria * Greek numerals
Greek numerals
in Unicode
Unicode
(acrophonic, not alphabetic, numerals) * Isopsephy * Number of the Beast

REFERENCES

* ^ _A_ _B_ Samuel Verdan (20 Mar 2007). "Systèmes numéraux en Grèce ancienne: description et mise en perspective historique" (in French). Retrieved 2 Mar 2011. * ^ _A_ _B_ _C_ Heath, Thomas L. _A Manual of Greek Mathematics_, pp. 14 ff. Oxford Univ. Press (Oxford), 1931. Reprinted Dover (Mineola ), 2003. Accessed 1 November 2013. * ^ Thompson, Edward M. _Handbook of Greek and Latin Palaeography_, p. 114. D. Appleton (New York), 1893. * ^ The Packard Humanities Institute (Cornell & Ohio State Universities). _Searchable Greek Inscriptions_: "IG I³ 1387" . Accessed 1 November 2013. * ^ Jeffery, Lilian H. _The Local Scripts of Archaic Greece_, pp. 38 ff. Clarendon (Oxford), 1961. * ^ The Packard Humanities Institute (Cornell & Ohio State Universities). _Searchable Greek Inscriptions_: "Magnesia 4" . Accessed 1 November 2013. * ^ The Packard Humanities Institute (Cornell & Ohio State Universities). _Searchable Greek Inscriptions_: "IG II² 2776". Accessed 1 November 2013. * ^ Edkins, Jo (2006). "Classical Greek Numbers". Retrieved 29 Apr 2013. * ^ Nick Nicholas (9 Apr 2005). "Numerals: Stigma, Koppa, Sampi". Retrieved 2 Mar 2011. * ^ Neugebauer, Otto (1969) . _The Exact Sciences in Antiquity_ (2 ed.). Dover Publications. pp. 13–14, plate 2. ISBN 978-0-486-22332-2 . * ^ Raymond Mercier, "Consideration of the Greek symbol \'zero\'" (PDF). (1.32 MiB ) Numerous examples * ^ _Ptolemy's Almagest
Almagest
_, translated by G. J. Toomer, Book VI, (Princeton, NJ: Princeton University Press, 1998), pp. 306–7.

EXTERNAL LINKS

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