Greek numerals
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Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, is a system of writing numbers using the letters of the
Greek alphabet The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC. It was derived from the earlier Phoenician alphabet, and is the earliest known alphabetic script to systematically write vowels as wel ...
. In modern
Greece Greece, officially the Hellenic Republic, is a country in Southeast Europe. Located on the southern tip of the Balkan peninsula, it shares land borders with Albania to the northwest, North Macedonia and Bulgaria to the north, and Turkey to th ...
, they are still used for
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s and in contexts similar to those in which
Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...
are still used in the
Western world The Western world, also known as the West, primarily refers to various nations and state (polity), states in Western Europe, Northern America, and Australasia; with some debate as to whether those in Eastern Europe and Latin America also const ...
. For ordinary
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s, however, modern Greece uses
Arabic numerals The ten Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) are the most commonly used symbols for writing numbers. The term often also implies a positional notation number with a decimal base, in particular when contrasted with Roman numera ...
.


History

The Minoan and Mycenaean civilizations' Linear A and
Linear B Linear B is a syllabary, syllabic script that was used for writing in Mycenaean Greek, the earliest Attested language, attested form of the Greek language. The script predates the Greek alphabet by several centuries, the earliest known examp ...
alphabets used a different system, called Aegean numerals, which included number-only symbols for powers of ten:  = 1,  = 10,  = 100,  = 1000, and  = 10000. Attic numerals composed another system that came into use perhaps in the 7th century BC. They were
acrophonic Acrophony (; + 'sound') is the naming of grapheme, letters of an alphabetic writing system so that a letter's name begins with the letter itself. For example, Greek letter names are acrophonic: the names of the letters α, β, γ, δ, are spell ...
, derived (after the initial one) from the first letters of the names of the numbers represented. They ran  = 1,  = 5,  = 10,  = 100,  = 1,000, and  = 10,000. The numbers 50, 500, 5,000, and 50,000 were represented by the letter with minuscule powers of ten written in the top-right corner: , , , and . One-half was represented by (left half of a full circle) and one-quarter by ɔ (right side of a full circle). The same system was used outside of
Attica Attica (, ''Attikḗ'' (Ancient Greek) or , or ), or the Attic Peninsula, is a historical region that encompasses the entire Athens metropolitan area, which consists of the city of Athens, the capital city, capital of Greece and the core cit ...
, but the symbols varied with the local alphabets; for example, 1,000 was in
Boeotia Boeotia ( ), sometimes Latinisation of names, Latinized as Boiotia or Beotia (; modern Greek, modern: ; ancient Greek, ancient: ), is one of the regional units of Greece. It is part of the modern regions of Greece, region of Central Greece (adm ...
. The present system probably developed around
Miletus Miletus (Ancient Greek: Μίλητος, Mílētos) was an influential ancient Greek city on the western coast of Anatolia, near the mouth of the Maeander River in present day Turkey. Renowned in antiquity for its wealth, maritime power, and ex ...
in
Ionia Ionia ( ) was an ancient region encompassing the central part of the western coast of Anatolia. It consisted of the northernmost territories of the Ionian League of Greek settlements. Never a unified state, it was named after the Ionians who ...
. 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 Athens ( ) is the Capital city, capital and List of cities and towns in Greece, largest city of Greece. A significant coastal urban area in the Mediterranean, Athens is also the capital of the Attica (region), Attica region and is the southe ...
abandoned its pre-Eucleidean alphabet in favour of
Miletus Miletus (Ancient Greek: Μίλητος, Mílētos) was an influential ancient Greek city on the western coast of Anatolia, near the mouth of the Maeander River in present day Turkey. Renowned in antiquity for its wealth, maritime power, and ex ...
's in 402 BC, and it may predate that by a century or two. The present system uses the 24 letters adopted under
Eucleides Eucleides () was eponymous archon of Athens for the year running from July/August 403 BC until June/July 402 BC. His year in office was marked by Athens' official adoption of the Ionic alphabet. There is some evidence that he may have been perso ...
, as well as three Phoenician and Ionic ones that had not been dropped from the Athenian alphabet (although kept for numbers):
digamma Digamma or wau (uppercase: Ϝ, lowercase: ϝ, numeral: ϛ) is an Archaic Greek alphabets, archaic letter of the Greek alphabet. It originally stood for the sound but it has remained in use principally as a Greek numeral for 6 (number), 6. Whe ...
, 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, and its use is unattested in Athens until the 2nd century CE. (In general, Athenians resisted using the new numerals for the longest of any Greek state, but had fully adopted them by .)


Description

Greek numerals are
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...
, based on powers of 10. The units from 1 to 9 are assigned to the first nine letters of the old
Ionic alphabet The history of the Greek alphabet starts with the adoption of Phoenician letter forms in the 9th–8th centuries BC during early Archaic Greece and continues to the present day. The Greek alphabet was developed during the Iron Age, centuries ...
from
alpha Alpha (uppercase , lowercase ) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter ''aleph'' , whose name comes from the West Semitic word for ' ...
to
theta Theta (, ) uppercase Θ or ; lowercase θ or ; ''thē̂ta'' ; Modern: ''thī́ta'' ) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth 𐤈. In the system of Greek numerals, it has a value of 9. Gree ...
. 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 Iota (; uppercase Ι, lowercase ι; ) is the ninth letter of the Greek alphabet. It was derived from the Phoenician letter Yodh. Letters that arose from this letter include the Latin I and J, the Cyrillic І (І, і), Yi (Ї, ї), and J ...
to koppa. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well, from
rho Rho (; uppercase Ρ, lowercase ρ or ; or ) is the seventeenth letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician alphabet, Phoenician letter resh . Its uppercase form uses the same ...
to sampi. (That this was not the traditional location of sampi in the Ionic alphabetical order has led classicists to conclude that sampi had fallen into disuse as a letter 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 Anatolia (), also known as Asia Minor, is a peninsula in West Asia that makes up the majority of the land area of Turkey. It is the westernmost protrusion of Asia and is geographically bounded by the Mediterranean Sea to the south, the Aegean ...
well into the Roman period.Heath, Thomas L. ''A Manual of Greek Mathematics''
pp. 14 ff.
Oxford Univ. Press (Oxford), 1931. Reprinted Dover ( Mineola), 2003. Accessed 1 November 2013.
) In ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars: , , , etc. In medieval manuscripts of the
Book of Revelation The Book of Revelation, also known as the Book of the Apocalypse or the Apocalypse of John, is the final book of the New Testament, and therefore the final book of the Bible#Christian Bible, Christian Bible. Written in Greek language, Greek, ...
, the
number of the Beast The number of the beast (, ) is associated with the The Beast (Revelation), Beast of Revelation in chapter 13, verse 18 of the Book of Revelation. In most manuscripts of the New Testament and in English translations of the Bible, the number of ...
666 is written as  (600 + 60 + 6). (Numbers larger than 1,000 reused the same letters but included various marks to note the change.) Fractions were indicated as the denominator followed by a ''keraia'' (ʹ); γʹ indicated one third, δʹ one fourth and so on. As an exception, special symbol ∠ʹ indicated one half, and γ°ʹ or γoʹ was two-thirds. These fractions were additive (also known as Egyptian fractions); for example indicated . Although the
Greek alphabet The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC. It was derived from the earlier Phoenician alphabet, and is the earliest known alphabetic script to systematically write vowels as wel ...
began with only
majuscule Letter case is the distinction between the letters that are in larger uppercase or capitals (more formally '' majuscule'') and smaller lowercase (more formally '' minuscule'') in the written representation of certain languages. The writing syste ...
forms, surviving
papyrus Papyrus ( ) is a material similar to thick paper that was used in ancient times as a writing surface. It was made from the pith of the papyrus plant, ''Cyperus papyrus'', a wetland sedge. ''Papyrus'' (plural: ''papyri'' or ''papyruses'') can a ...
manuscripts from
Egypt Egypt ( , ), officially the Arab Republic of Egypt, is a country spanning the Northeast Africa, northeast corner of Africa and Western Asia, southwest corner of Asia via the Sinai Peninsula. It is bordered by the Mediterranean Sea to northe ...
show that uncial and
cursive Cursive (also known as joined-up writing) is any style of penmanship in which characters are written joined in a flowing manner, generally for the purpose of making writing faster, in contrast to block letters. It varies in functionality and m ...
minuscule Letter case is the distinction between the letters that are in larger uppercase or capitals (more formally ''majuscule'') and smaller lowercase (more formally '' minuscule'') in the written representation of certain languages. The writing system ...
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 (Ϝ) came to be avoided in favour 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 Sigma ( ; uppercase Σ, lowercase σ, lowercase in word-final position ς; ) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as an operator ...
-
tau Tau (; uppercase Τ, lowercase τ or \boldsymbol\tau; ) is the nineteenth letter of the Greek alphabet, representing the voiceless alveolar plosive, voiceless dental or alveolar plosive . In the system of Greek numerals, it has a value of 300 ...
ligature stigma ϛ ( or ). In
modern Greek Modern Greek (, or , ), generally referred to by speakers simply as Greek (, ), refers collectively to the dialects of the Greek language spoken in the modern era, including the official standardized form of the language sometimes referred to ...
, a number of other changes have been made. Instead of extending an over bar over an entire number, the ''keraia'' (, 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 similar to the
acute accent The acute accent (), , is a diacritic used in many modern written languages with alphabets based on the Latin alphabet, Latin, Cyrillic script, Cyrillic, and Greek alphabet, Greek scripts. For the most commonly encountered uses of the accen ...
(´), the tonos (U+0384,΄) and the prime symbol (U+02B9, ʹ), but has its own
Unicode Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Char ...
character as U+0374.
Alexander the Great Alexander III of Macedon (; 20/21 July 356 BC – 10/11 June 323 BC), most commonly known as Alexander the Great, was a king of the Ancient Greece, ancient Greek kingdom of Macedonia (ancient kingdom), Macedon. He succeeded his father Philip ...
's father
Philip II of Macedon Philip II of Macedon (; 382 BC – October 336 BC) was the king (''basileus'') of the ancient kingdom of Macedonia (ancient kingdom), Macedonia from 359 BC until his death in 336 BC. He was a member of the Argead dynasty, founders of the ...
is thus known as in modern Greek. A lower left ''keraia'' (Unicode: U+0375, "Greek Lower Numeral Sign") is now standard for distinguishing thousands: 2019 is represented as ͵ΒΙΘʹ (). 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.


Isopsephy

The practice of adding up the number values of Greek letters of words, names and phrases, thus connecting the meaning of words, names and phrases with others with equivalent numeric sums, is called ''
isopsephy In numerology, isopsephy (stressed on the ''I'' and the ''E''; , ) or isopsephism is the practice of adding up the Greek numerals, number values of the letters in a word to form a single number. The total number is then used as a metaphorical brid ...
''. Similar practices for the Hebrew and English are called ''
gematria In numerology, gematria (; or , plural or ) is the practice of assigning a numerical value to a name, word, or phrase by reading it as a number, or sometimes by using an alphanumeric cipher. The letters of the alphabets involved have standar ...
'' and English Qaballa, respectively.


Table

* Alternatively, sub-sections of manuscripts are sometimes numbered by lowercase characters (αʹ. βʹ. γʹ. δʹ. εʹ. ϛʹ. ζʹ. ηʹ. θʹ.). * In Ancient Greek, myriad notation is used for multiples of 10,000, for example for 20,000 or (also written on the line as Μ ) for 1,234,567.Greek number systems - MacTutor
/ref>


Higher numbers

In his text '' The Sand Reckoner'', the natural philosopher
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
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.
Pappus of Alexandria Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...
reports that
Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...
developed a simpler system based on powers of the myriad; was 10,000, was 10,0002 = 100,000,000, was 10,0003 = 1012 and so on.


Zero

Hellenistic In classical antiquity, the Hellenistic period covers the time in Greek history after Classical Greece, between the death of Alexander the Great in 323 BC and the death of Cleopatra VII in 30 BC, which was followed by the ascendancy of the R ...
astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
s extended alphabetic Greek numerals into a
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
positional numbering system by limiting each position to a maximum value of 50 + 9 and including a special symbol for
zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
, which was only used alone for a whole table cell, rather than combined with other digits, like today's modern zero, which is a placeholder in positional numeric notation. This system was probably adapted from Babylonian numerals by
Hipparchus Hipparchus (; , ;  BC) was a Ancient Greek astronomy, Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hippar ...
. It was then used by
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
(), Theon () and Theon's daughter
Hypatia Hypatia (born 350–370 – March 415 AD) was a Neoplatonist philosopher, astronomer, and mathematician who lived in Alexandria, Egypt (Roman province), Egypt: at that time a major city of the Eastern Roman Empire. In Alexandria, Hypatia was ...
(). The symbol for zero is clearly different from that of the value for 70,
omicron Omicron (, ; uppercase Ο, lowercase ο, ) is the fifteenth letter of the Greek alphabet. This letter is derived from the Phoenician letter ayin: . In classical Greek, omicron represented the close-mid back rounded vowel in contrast to '' o ...
or " ο". In the 2nd-century papyrus shown here, one can see the symbol for zero in the lower right, and a number of larger omicrons elsewhere in the same papyrus. In
Ptolemy's table of chords The table of chords, created by the Greece, Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's ''Almagest'', a treatise on mathematical astron ...
, the first fairly extensive trigonometric table, there were 360 rows, portions of which looked as follows: : \begin \pi\varepsilon\varrho\iota\varphi\varepsilon\varrho\varepsilon\iota\tilde\omega\nu & \varepsilon\overset\upsilon\vartheta\varepsilon\iota\tilde\omega\nu & \overset\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\ \begin \hline \pi\delta\angle' \\ \pi\varepsilon \\ \pi\varepsilon\angle' \\ \hline \pi\stigma \\ \pi\stigma\angle' \\ \pi\zeta \\ \hline \end & \begin \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\stigma & \vartheta \\ \hline \end & \begin \hline \circ & \circ & \mu\stigma & \kappa\varepsilon \\ \circ & \circ & \mu\stigma & \iota\delta \\ \circ & \circ & \mu\stigma & \gamma \\ \hline \circ & \circ & \mu\varepsilon & \nu\beta \\ \circ & \circ & \mu\varepsilon & \mu \\ \circ & \circ & \mu\varepsilon & \kappa\vartheta \\ \hline \end \end Each number in the first column, labeled regions"is the number of degrees of arc on a circle. Each number in the second column, labeled straight lines" or "segments"is the length of the corresponding chord of the circle, when the diameter is 120. Thus represents an 84° arc, and the after it means one-half, so that means °. In the next column we see  , meaning . That is the length of the chord corresponding to an arc of ° when the diameter of the circle is 120. The next column, labeled 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 1°. Thus that last column was used for
linear interpolation In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known po ...
. The Greek
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
placeholder or zero symbol changed over time: The symbol used on
papyri Papyrus ( ) is a material similar to thick paper that was used in ancient times as a writing surface. It was made from the pith of the papyrus plant, ''Cyperus papyrus'', a wetland sedge. ''Papyrus'' (plural: ''papyri'' or ''papyruses'') can ...
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, later the overbar was omitted in
Byzantine The Byzantine Empire, also known as the Eastern Roman Empire, was the continuation of the Roman Empire centred on Constantinople during late antiquity and the Middle Ages. Having survived the events that caused the fall of the Western Roman E ...
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 meaning "nothing". 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
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
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 The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
and either the center of the Sun (for
solar eclipse A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby obscuring the view of the Sun from a small part of Earth, totally or partially. Such an alignment occurs approximately every six months, during the eclipse season i ...
s) or the center of
Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
's shadow (for
lunar eclipse A lunar eclipse is an astronomical event that occurs when the Moon moves into the Earth's shadow, causing the Moon to be darkened. Such an alignment occurs during an eclipse season, approximately every six months, during the full moon phase, ...
s). All of these zeros took the form , 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. The Greek zero was added to Unicode in Version 4.1.0 at .


See also

* * * * * (acrophonic, not alphabetic, numerals) * , based on the Greek system * * * * * * * *


References


External links


The Greek Number Converter
{{DEFAULTSORT:Greek Numerals Numeral systems Numerals Numerals