Greek numerals

Numeral systems

Hindu–Arabic numeral system
Western Arabic Eastern Arabic Bengali Gurmukhi Indian Sinhala Tamil Balinese Burmese Dzongkha Gujarati Javanese Khmer Lao Mongolian Thai
East Asian
Chinese Suzhou Hokkien Japanese Korean Vietnamese Counting rods
Alphabetic
Abjad Armenian Āryabhaṭa Cyrillic Ge'ez Georgian Greek Hebrew Roman
Former
Aegean Attic Babylonian Brahmi Egyptian Etruscan Inuit Kharosthi Mayan Muisca Quipu Prehistoric
Positional systems by base
2 3 4 5 6 8 10 12 16 20 60
Non-standard positional numeral systems
Bijective numeration (1) Signed-digit representation (Balanced ternary) factorial negative Complex-base system (2i) Non-integer representation (φ) mixed
List of numeral systems

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

History

The Minoan and Mycenaean civilizations' 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.^[1]

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 .^[1] The same system was used outside of Attica, but the symbols varied with the local alphabets: in Boeotia, was 1000.^[2]

The present system probably developed around Miletus in Ionia. 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use.^[3] More thorough modern archaeology has caused the date to be pushed back at least to the 5th century BC,^[4] a little before Athens abandoned its pre-Euclidean alphabet in favor of Miletus's in 402 BC, and it may predate that by a century or two.^[5] The present system uses the 24 letters adopted by 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 does not appear until the 2nd century BC^[6] and its use is unattested in Athens until the 2nd century AD.^[7] (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.^[2])

Description

Greek numerals in a c. 1100 Byzantine manuscript of Hero of 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's Geography, using Greek numerals for its graticule: 52–63°N of the equator and 6–33°E from Ptolemy's Prime Meridian at the Fortunate Isles.

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 alphabet from 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.^[8] (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 well into the Roman period.^[2]) 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, 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 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 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: 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.^[9]

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

Byzantine	Modern	Value	Byzantine	Modern	Value	Ancient	Byzantine	Modern	Value	Ancient	Byzantine	Modern	Value
α	Αʹ	1	ι	Ιʹ	10		ρ	Ρʹ	100	&	͵α	͵Α	1000
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
& &	Ϛʹ ΣΤʹ	6	ξ	Ξʹ	60		χ	Χʹ	600		͵ & ͵ ͵ & ͵	͵Ϛ	6000
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	& &	& & &	Ϡʹ	900		͵θ	͵Θ	9000

Alternatively, sub-sections of manuscripts are sometimes numbered {1,2,3,...,9} by lowercase characters {αʹ. βʹ. γʹ. δʹ. εʹ. ϛʹ. ζʹ. ηʹ. θʹ.}
In Ancient Greek, Myriad notation is used for numbers larger than 9,999, e.g. ${\stackrel {\rho o\epsilon }{\mathrm {M} }}$ ͵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 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 by Hipparchus c. 140 BC. It was then used by Ptolemy (c. 140), Theon (c. 380) and Theon's daughter 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:

{\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}{|r|r|r|}\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}{|r|r|r|r|}\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 &\kappa \vartheta \\\hline \end{array}}\end{array}}

Each number in the first column, labeled περιφερειῶν, is the number of degrees of arc on a circle. Each number in the second column, labeled ευθειῶν, is the length of the corresponding chord of the circle, when the diameter is 120. Thus πδ represents an 84° arc, and the ∠' after it means one-half, so that πδ∠' means 84.5°. In the next column we see π μα γ, meaning 80 + 41/60 + 3/60². That is the length of the chord corresponding to an arc of 84.5° when the diameter of the circle is 120. The next column, labeled ὲξηκοστῶν, for "sixtieths", is the number to be added to the chord length for each 1° increase in the arc, over the span of the next 12°. Thus that last column was used for linear interpolation.

The Greek sexagesimal placeholder 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 the 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".^[10]^[11] 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 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 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.^[12]

References

1 2 Samuel Verdan (20 Mar 2007). "Systèmes numéraux en Grèce ancienne: description et mise en perspective historique" (in French). Retrieved 2 Mar 2011.
1 2 3 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" [also known as IG I² 760]. 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" [also known as Syll³ 695.b]. 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) [1957]. 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, translated by G. J. Toomer, Book VI, (Princeton, NJ: Princeton University Press, 1998), pp. 306–7.

External links

Wikimedia Commons has media related to Greek numerals.

The Greek Number Converter

Greek language

Origin and genealogy

Periods

Mycenaean Greek (c. 1600–1100 BC)
Ancient Greek (c. 800–300 BC)
Koine Greek (c. 300 BC–AD 330)
Jewish Koine Greek
Medieval Greek (c. 330–1453)
Modern Greek (since 1453)

Varieties

Ancient	Aeolic Arcadocypriot Attic and Ionic Doric Homeric Locrian Pamphylian Macedonian
Modern	Cappadocian Misthiotika Cretan Cypriot Demotic Himariote Italiot Greco/Calabrian Griko/Apulian Katharevousa Maniot Mariupolitan Pontic Tsakonian Yevanic

Phonology

Grammar

Writing systems

Literature

Promotion and study

Other

Ancient Greece

Periods	Cycladic civilization Minoan civilization Mycenaean civilization Greek Dark Ages Archaic period Classical Greece Hellenistic Greece Roman Greece
Geography	Aegean Sea Aeolis Alexandria Antioch Cappadocia Crete Cyprus Doris Ephesus Epirus Hellespont Ionia Ionian Sea Macedonia Magna Graecia Miletus Peloponnesus Pergamon Pontus Taurica Ancient Greek colonies

City states

Politics

Boeotarch Boule Koinon Proxeny Strategos Tagus Tyrant Amphictyonic League
Athenian	Agora Areopagus Ecclesia Graphē paranómōn Heliaia Ostracism
Spartan	Apella Ephor Gerousia Harmost
Macedon	Synedrion Koinon

Military

People

List of ancient Greeks
Rulers	Kings of Argos Archons of Athens Kings of Athens Kings of Commagene Diadochi Kings of Lydia Kings of Macedonia Kings of Paionia Attalid kings of Pergamon Kings of Pontus Kings of Sparta Tyrants of Syracuse
Philosophers	Anaxagoras Anaximander Anaximenes Antisthenes Aristotle Democritus Diogenes of Sinope Empedocles Epicurus Gorgias Heraclitus Hypatia Leucippus Parmenides Plato Protagoras Pythagoras Socrates Thales Zeno
Authors	Aeschylus Aesop Alcaeus Archilochus Aristophanes Bacchylides Euripides Herodotus Hesiod Hipponax Homer Ibycus Lucian Menander Mimnermus Panyassis Philocles Pindar Plutarch Polybius Sappho Simonides Sophocles Stesichorus Theognis Thucydides Timocreon Tyrtaeus Xenophon
Others	Agesilaus II Agis II Alcibiades Alexander the Great Aratus Archimedes Aspasia Demosthenes Epaminondas Euclid Hipparchus Hippocrates Leonidas Lycurgus Lysander Milo of Croton Miltiades Pausanias Pericles Philip of Macedon Philopoemen Praxiteles Ptolemy Pyrrhus Solon Themistocles
Groups	Philosophers Playwrights Poets Tyrants
By culture	Ancient Greek tribes Thracian Greeks Ancient Macedonians

Society

Arts
Sciences

Architecture (Greek Revival architecture)
Astronomy
Literature
Mathematics
Medicine
Music (Musical system)
Pottery
Sculpture
Technology
Theatre

Religion

Funeral and burial practices Mythology mythological figures Temple Twelve Olympians Underworld
Sacred places	Eleusis Delphi Delos Dodona Mount Olympus Olympia

Structures

Athenian Treasury Lion Gate Long Walls Philippeion Theatre of Dionysus Tunnel of Eupalinos
Temples	Aphaea Artemis Athena Nike Erechtheion Hephaestus Hera (Olympia) Parthenon Samothrace Zeus (Olympia)

Language

Proto-Greek
Mycenaean
Homeric
Dialects
- Aeolic
- Arcadocypriot
- Attic
- Doric
- Ionic
- Locrian
- Macedonian
- Pamphylian
Koine

Writing

Lists
Cities in Epirus People Place names Stoae Temples Theatres

Category
Portal

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.