TheInfoList

Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are 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 ninth or early eighth century BC. It is derived from the earlier Phoenician alphabet, and was the first alphabetic script in history to have distinct letters for vowels ...

. In modern
Greece Greece ( el, Ελλάδα, Elláda, ), officially the Hellenic Republic, is a country located in Southeastern Europe Southeast Europe or Southeastern Europe () is a geographical subregion A subregion is a part of a larger region In geogr ...

, they are still used for
ordinal number In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a s ...
s and in contexts similar to those in which
Roman numerals Roman numerals are a that originated in and remained the usual way of writing numbers throughout Europe well into the . Numbers in this system are represented by combinations of letters from the . Modern style uses seven symbols, each with a ...
are still used in the
Western world The Western world, also known as the West, refers to various regions, nations and state (polity), states, depending on the context, most often consisting of the majority of Europe, Northern America, and Australasia.
. For ordinary
cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
s, however, modern Greece uses
Arabic numerals Arabic numerals are the ten numerical digit A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a Positional notation, positional numeral sy ...

.

# History

The
Minoan The Minoan civilization was a Bronze Age Aegean civilization on the island of Crete and other Aegean Islands, flourishing from c. 3000 BC to c. 1450 BC and, after a late period of decline, finally ending around 1100 BC, during the early Greek Da ...
and
Mycenaean civilization Mycenaean Greece (or the Mycenaean civilization) was the last phase of the Bronze Age in Ancient Greece, spanning the period from approximately 1750 to 1050 BC.. It represents the first advanced and distinctively Greek civilization in mainland ...
s'
Linear A Linear A is a writing system that was used by the (Cretans) from 1800 to 1450 BC to write the hypothesized . Linear A was the primary script used in palace and religious writings of the Minoan civilization. It was discovered by arch ...
and
Linear B Linear B is a syllabic script that was used for writing Mycenaean Greek, the earliest attested form of Greek. The script predates the Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or ...
alphabets used a different system, called
Aegean numerals Aegean may refer to: *Aegean Sea The Aegean Sea ; tr, Ege Denizi is an elongated Bay, embayment of the Mediterranean Sea located between Europe's Geography of Europe, Balkan peninsula and Asia's Anatolia peninsula. The sea has an area of s ...
, which included number-only symbols for powers of ten:  = 1,  = 10,  = 100,  = 1000, and  = 10000.
Attic numerals File:Aegina Prison Attic Numerals.jpg, Plaque above the main entrance to the orphanage, which later became a prison, on the Greek island of Aegina. The ancient Greek inscription translates as “The Governor erected this orphanage in the year 18 ...
comprised another system that came into use perhaps in the 7th century . They were
acrophonic Acrophony (; Greek: ἄκρος ''akros'' uppermost + φωνή ''phone'' 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: ...
, 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-quarter was represented by (right half of a full circle) and one-half by the left side of the circle. The same system was used outside of
Attica Attica ( el, Αττική, Ancient Greek ''Attikḗ'' or , or ), or the Attic peninsula, is a historical region that encompasses the city of Athens, the capital city, capital of Greece and its countryside. It is a peninsula projecting into the ...

, but the symbols varied with the local alphabets, for example, 1,000 was in
Boeotia Boeotia, sometimes alternatively Latinised Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920 ...

. The present system probably developed around
Miletus Miletus (; gr, Μῑ́λητος, Mīlētos; Hittite language, Hittite transcription ''Millawanda'' or ''Milawata'' (Exonym and endonym, exonyms); la, Miletus; tr, Milet) was an Ancient Greece, ancient Greek city on the western coast of Ana ...
in
Ionia Ionia (; Ancient Greek Ancient Greek includes the forms of the used in and the from around 1500 BC to 300 BC. It is often roughly divided into the following periods: (), Dark Ages (), the period (), and the period (). Ancient ...
. 19th century classicists placed its development in the 3rd century , the occasion of its first widespread use. More thorough
modern archaeology Modern archaeology is the discipline of archaeology which contributes to excavations. Johann Joachim Winckelmann was one of the founders of scientific archaeology and first applied the categories of style on a large, systematic basis to the h ...
has caused the date to be pushed back at least to the 5th century , a little before
Athens , image_skyline = File:Athens Montage L.png, center, 275px, alt=Athens montage. Clicking on an image in the picture causes the browser to load the appropriate article. rect 15 15 985 460 Acropolis of Athens rect 15 475 48 ...
abandoned its
pre-Euclidean alphabet Many local variants of the Greek alphabet were employed in ancient Greece during the Archaic period in Greece, archaic and Classical period in Greece, early classical periods, until they were replaced by the classical 24-letter alphabet that is the ...
in favour of
Miletus Miletus (; gr, Μῑ́λητος, Mīlētos; Hittite language, Hittite transcription ''Millawanda'' or ''Milawata'' (Exonym and endonym, exonyms); la, Miletus; tr, Milet) was an Ancient Greece, ancient Greek city on the western coast of Ana ...
's in 402 , and it may predate that by a century or two. The present system uses the 24 letters used by
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...
, as well as three
Phoenician Phoenician may refer to: * Phoenicia, an ancient civilization * Phoenician alphabet * Phoenician language * List of Phoenician cities * Phoenix, Arizona See also

* Phoenix (mythology) * Phoenicia (disambiguation) {{disambiguation Language an ...

and Ionic ones that had not been dropped from the Athenian alphabet (although kept for numbers):
digamma Digamma, waw, or wau (uppercase: Ϝ, lowercase: ϝ, numeral: ϛ) is an archaic letter of the Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the ...
, koppa, and
sampi Sampi (modern: ϡ; ancient shapes: , ) is an archaic letter of the Greek alphabet. It was used as an addition to the classical 24-letter alphabet in some eastern Ionic Greek, Ionic dialects of ancient Greek in the 6th and 5th centuries BC, to de ...
. 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 Sampi (modern: ϡ; ancient shapes: , ) is an archaic letter of the Greek alphabet. It was used as an addition to the classical 24-letter alphabet in some eastern Ionic Greek, Ionic dialects of ancient Greek in the 6th and 5th centuries BC, to de ...

, is problematic since its uncommon value means the first attested representative near Miletus does not appear until the 2nd century , and its use is unattested in Athens until the 2nd century . (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 A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
, based on powers of 10. The units from 1 to 9 are assigned to the first nine letters of the old
Ionic alphabet vessel with double alphabet inscription, showing new letters ΥΧ , and ΥΧΦΨΩ , 450–425 BC The history of the Greek alphabet starts with the adoption of Phoenician alphabet, Phoenician letter forms and continues to the present day. The ...
from
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', modern pronunciation ''álfa'') is the first letter Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet) A letter is a segmental symbol A s ...
to
theta Theta (, ; uppercase Θ or ϴ, lowercase θ or ϑ; grc, ''thē̂ta'' ; Modern Modern may refer to: History *Modern history Human history, also known as world history, is the description of humanity's past. It is informed by archaeo ...
. 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 The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician alpha ...
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 ; el, ῥῶ) is the 17th letter of the Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician ...
to
sampi Sampi (modern: ϡ; ancient shapes: , ) is an archaic letter of the Greek alphabet. It was used as an addition to the classical 24-letter alphabet in some eastern Ionic Greek, Ionic dialects of ancient Greek in the 6th and 5th centuries BC, to de ...
. (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,, tr, Anadolu Yarımadası), and the Anatolian plateau. also known as Asia Minor, is a large peninsula A peninsula ( la, paeninsula from 'almost' and 'island') is a landform A landform is a natural or artificial feature of ...

well into the
Roman period The Roman Empire ( la, Imperium Rōmānum ; grc-gre, Βασιλεία τῶν Ῥωμαίων, Basileía tôn Rhōmaíōn) was the post-Republican Republican can refer to: Political ideology * An advocate of a republic, a type of governme ...
.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
overbar An overline, overscore, or overbar, is a typographical feature of a horizontal and vertical, horizontal line drawn immediately above the text. In old mathematical notation, an overline was called a ''vinculum (symbol), vinculum'', a notation fo ...
s: , , , etc. In medieval manuscripts of the
Book of Revelation The Book of Revelation (also called the Apocalypse of John, Revelation to John or Revelation from Jesus Christ) is the final book of the New Testament The New Testament grc, Ἡ Καινὴ Διαθήκη, Transliteration, transl. ; ...
, the
number of the BeastThe number of the beast, 666 in most manuscripts, is associated with the Beast of Revelation in the Book of Revelation in the Bible. Number of the Beast also may refer to: * ''The Number of the Beast'' (album), 1982, by Iron Maiden ** "The Numbe ...
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 fraction An Egyptian fraction is a finite sum of distinct unit fractionA unit fraction is a rational number written as a fraction where the numerator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, ...
s); for example indicated . Although the
Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician alphabet, and was the first alphabetic script in history to have distinct letters for vowels ...
began with only
majuscule Letter case is the distinction between the letters Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet) A letter is a segmental symbol A symbol is a mark, sign, or word that indicates, signifies, or ...
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'') can also refer to a do ...

manuscripts from
Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a transcontinental country This is a list of countries located on more than one continent A continent is one of several large landmasses. Generally identi ...

show that
uncial Uncial is a majuscule Letter case is the distinction between the letters Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet) A letter is a segmental symbol A symbol is a mark, sign, or word that ...

and
cursive Cursive (also known as script, among other names) is any style of penmanship Penmanship is the technique of writing Writing is a medium of human communication that involves the representation of a language with written symbols. Writi ...

minuscule Letter case (or just case) is the distinction between the Letter (alphabet), letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation ...
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 Byzantine calendar, also called "Creation Era of Constantinople" or "Era of the World" ( grc, Ἔτη Γενέσεως Κόσμου κατὰ Ῥωμαίους, also or , abbreviated as ε.Κ.; literal translation of ancient Greek "Roman ye ...
, the letter was known as episemon and written as or . This eventually merged with the
sigma Sigma (uppercase Letter case is the distinction between the letters Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet) A letter is a segmental symbol A symbol is a mark, sign, or word that ind ...
-
tau Tau (uppercase Τ, lowercase τ; el, ταυ ) is the 19th letter of the Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician ...
ligature Ligature may refer to: * Ligature (medicine), a piece of suture used to shut off a blood vessel or other anatomical structure ** Ligature (orthodontic), used in dentistry * Ligature (music), an element of musical notation used especially in the med ...
stigma Stigma or plural stigmata, stigmas may refer to: * Social stigma, the disapproval of a person based on physical or behavioral characteristics that distinguish them from others Symbolism * Stigmata, bodily marks or wounds resembling the crucifix ...
ϛ ( or ). In
modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialect The term dialect (from Latin , , from the Ancient Greek word , 'discourse', from , 'through' ...
, 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 A diacritic (also diacritical mark, diacritical point, diacritical sign, or accent) is a glyph The term glyph is used in typography File:metal movable type.jpg, 225px, Movable type being assembl ...

(´), the
tonos Greek orthography has used a variety of diacritics starting in the Hellenistic period. The more complex polytonic orthography ( el, πολυτονικό σύστημα γραφής, translit=polytonikó sýstīma grafīs), which includes five d ...

(U+0384,΄) and the prime symbol (U+02B9, ʹ), but has its own
Unicode Unicode, formally the Unicode Standard, is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's wri ...

character as U+0374.
Alexander the Great Alexander III of Macedon ( grc-gre, Αλέξανδρος}, ; 20/21 July 356 BC – 10/11 June 323 BC), commonly known as Alexander the Great, was a king (''basileus ''Basileus'' ( el, βασιλεύς) is a Greek term and title A title ...

's father
Philip II of Macedon Philip II of Macedon ( grc-gre, Φίλιππος ; 382 – 21 October 336 BC) was the king (basileus ''Basileus'' ( el, βασιλεύς) is a Greek term and title A title is one or more words used before or after a person's name, in cer ...
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 (gematria)

The art of assigning Greek letters also being thought of as numerals and therefore giving words, names and phrases a numeric sum that has meaning through being connected to words, names and phrases of similar sum is called
isopsephy Isopsephy (; ''isos'' meaning "equal" and ''psephos'' meaning "pebble") 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 ...
(
gematria Gematria (; he, גמטריא or gimatria , plural or , ''gimatriot'') is the practice of assigning a numerical value A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, ...
). No

# 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 A myriad (from Ancient Greek Ancient Greek includes the forms of the Greek language Greek ( el, label=Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), ...
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 Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, ...
'', the natural philosopher
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

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 (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity, known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem ...
reports that
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος ''Apollonios o Pergeos''; la, Apollonius Pergaeus; ) was an Ancient Greece, Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the ...
developed a simpler system based on
powers Powers (stylized as POWERS) is a musical duo composed of Mike Del Rio and Crista Ru. Their music has been described as alternative pop, electropop, and Progressive pop, progressive pop. ''Time'' has called their music "groovy and futuristic". ...
of the myriad; was 10,000, was 10,0002 = 100,000,000, was 10,0003 = 1012 and so on.

# Zero

Hellenistic The Hellenistic period spans the period of History of the Mediterranean region, Mediterranean history between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in 31  ...
astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...

s extended alphabetic Greek numerals into a
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
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 A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

, 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 450px, Babylonian cuneiform numerals Assyro-Chaldean Babylonian cuneiform numerals were written in cuneiform Cuneiform is a Logogram, logo-Syllabary, syllabic writing system, script that was used to write several languages of the Ancient Nea ...

by
Hipparchus Hipparchus of Nicaea (; el, Ἵππαρχος, ''Hipparkhos'';  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 discov ...
. It was then used by
Ptolemy Claudius Ptolemy (; grc-koi, Κλαύδιος Πτολεμαῖος, , ; la, Claudius Ptolemaeus; AD) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes ...
(), Theon () and Theon's daughter
Hypatia Hypatia, Koine Greek, Koine pronunciation (born 350–370; died 415 AD) was a Greeks, Greek Neoplatonism, Neoplatonist philosopher, astronomer, and mathematician, who lived in Alexandria, Egypt (Roman province), Egypt, then part of the East ...
(died 415). The symbol for zero is clearly different from that of the value for 70,
omicron Omicron (; uppercase Ο, lowercase ο, ) is the 15th letter of the Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is derived from the earlier Phoenician alphab ...

or "". 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 chordsThe 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 astronom ...
, the first fairly extensive trigonometric table, there were 360 rows, portions of which looked as follows: : $\begin \pi\varepsilon\varrho\iota\varphi\varepsilon\varrho\varepsilon\iota\tilde\omega\nu & \overset\varepsilon\nu\vartheta\varepsilon\iota\tilde\omega\nu & \overset\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\ \begin \hline \pi\delta\angle\text{'} \\ \pi\varepsilon \\ \pi\varepsilon\angle\text{'} \\ \hline \pi\stigma \\ \pi\stigma\angle\text{'} \\ \pi\zeta \\ \hline \end & \begin \hline \pi & \mu\alpha & \gamma \\ \pi\alpha & \delta & \iota\varepsilon \\ \pi\alpha & \kappa\zeta & \kappa\beta \\ \hline \pi\alpha & \nu & \kappa\delta \\ \pi\beta & \iota\gamma & \iota\vartheta \\ \pi\beta & \lambda\stigma & \vartheta \\ \hline \end & \begin \hline \circ & \circ & \mu\stigma & \kappa\varepsilon \\ \circ & \circ & \mu\stigma & \iota\delta \\ \circ & \circ & \mu\stigma & \gamma \\ \hline \circ & \circ & \mu\varepsilon & \nu\beta \\ \circ & \circ & \mu\varepsilon & \mu \\ \circ & \circ & \mu\varepsilon & \kappa\vartheta \\ \hline \end \end$ Each number in the first column, labeled is the number of degrees of arc on a circle. Each number in the second column, labeled 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 πδ∠′ 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 12°. Thus that last column was used for
linear interpolation In mathematics, linear interpolation is a method of curve fitting Curve fitting is the process of constructing a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but ...

. The Greek
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
placeholder or zero symbol changed over time: The symbol used on
papyri Papyrus ( ) is a material similar to thick paper Paper is a thin sheet material produced by mechanically or chemically processing cellulose fibres derived from wood, Textile, rags, poaceae, grasses or other vegetable sources in water, dr ...

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 The Byzantine Empire, also referred to as the Eastern Roman Empire, or Byzantium, was the continuation of the Roman Empire in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinople. It survi ...

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 or sexagenary, is a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
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. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

and either the center of the
Sun The Sun is the star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many othe ...

(for
solar eclipse A solar eclipse occurs when a portion of the Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The r ...

s) or the center of
Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

lunar eclipse A lunar eclipse occurs when the Moon The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Sola ...

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.

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Attic numerals File:Aegina Prison Attic Numerals.jpg, Plaque above the main entrance to the orphanage, which later became a prison, on the Greek island of Aegina. The ancient Greek inscription translates as “The Governor erected this orphanage in the year 18 ...
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Cyrillic numerals Cyrillic numerals are a numeral system derived from the Cyrillic script, developed in the First Bulgarian Empire in the late 10th century. It was used in the First Bulgarian Empire and by South Slavs, South and East Slavs, East Slavic peoples. Th ...
* Greek numerals in Unicode (acrophonic, not alphabetic, numerals) *
Isopsephy Isopsephy (; ''isos'' meaning "equal" and ''psephos'' meaning "pebble") 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 ...
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Number of the BeastThe number of the beast, 666 in most manuscripts, is associated with the Beast of Revelation in the Book of Revelation in the Bible. Number of the Beast also may refer to: * ''The Number of the Beast'' (album), 1982, by Iron Maiden ** "The Numbe ...
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Alphabetic numeral system An alphabetic numeral system is a type of numeral system. Developed in classical antiquity, it flourished during the early Middle Ages. In alphabetic numeral systems, numbers are written using the Grapheme, characters of an alphabet, syllabary, or ...