View source for Greek numerals

{{short description|System of writing numbers}}
{{hatnote|The numerical signs ʹ<!-- U+02B9 --> and ͵ redirect here. For the accent mark ', see Acute accent.}}
{{numeral systems}}
'''Greek numerals''', also known as '''Ionic''', '''Ionian''', '''Milesian''', or '''Alexandrian numerals''', are a [[Numeral system|system of writing numbers]] using the letters of the Greek alphabet. In modern Greece, they are still used for [[Philosophy:Ordinal number (linguistics)|ordinal number]]s and in contexts similar to those in which [[Roman numerals]] are still used in the Western world. For ordinary [[Philosophy:Cardinal number (linguistics)|cardinal number]]s, however, modern Greece uses [[Arabic numerals]].

==History==
The Minoan and Mycenaean civilizations' [[Social:Linear A|Linear A]] and [[Social:Linear B|Linear B]] alphabets used a different system, called [[Aegean numerals]], which included number-only symbols for powers of ten: {{lang|gmy|𐄇}}&nbsp;=&nbsp;1, {{lang|gmy|𐄐}}&nbsp;=&nbsp;10, {{lang|gmy|𐄙}}&nbsp;=&nbsp;100, {{lang|gmy|𐄢}}&nbsp;=&nbsp;1000, and {{lang|gmy|𐄫}}&nbsp;=&nbsp;10000.<ref name=Verdun-2007-03-20>{{cite web |last=Verdan |first=Samuel |date=20 March 2007 |title=Systèmes numéraux en Grèce ancienne: Description et mise en perspective historique |lang=fr |url=http://www.dma.ens.fr/culturemath/histoire%20des%20maths/htm/Verdan/Verdan.htm |access-date=2 March 2011 |archive-date=2 February 2010 |archive-url=https://web.archive.org/web/20100202054101/http://www.dma.ens.fr/culturemath/histoire%20des%20maths/htm/Verdan/Verdan.htm |url-status=dead }}</ref>

[[Attic numerals]] composed another system that came into use perhaps in the 7th&nbsp;century&nbsp;BCE. They were acrophonic, derived (after the initial one) from the first letters of the names of the numbers represented. They ran {{GrGl|Zeta archaic}}&nbsp;=&nbsp;1, {{GrGl|Pi archaic}}&nbsp;=&nbsp;5, {{GrGl|Delta 04}}&nbsp;=&nbsp;10, {{GrGl|Eta classical}}&nbsp;=&nbsp;100, {{GrGl|Chi normal}}&nbsp;=&nbsp;1,000, and {{GrGl|Mu classical}}&nbsp;=&nbsp;10,000. The numbers 50, 500, 5,000, and 50,000 were represented by the letter {{GrGl|Pi archaic}} with minuscule powers of ten written in the top right corner: [[File:Attic_00050.svg|x16px]], x16px, x16px, and x16px.<ref name=Verdun-2007-03-20/> One-half was represented by {{script|Grek|𐅁}}<!-- can also use {{big|<math>\boldsymbol\mathsf\subset</math>}}, not browser-font dependent; really should be sans-serif, capital Lunate sigma Ϲ, or non-existant TeX <math>\boldsymbol\mathsf\Varsigma</math> (vs. l.c. \varsigma) --> (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, but the symbols varied with the local alphabets, for example, 1,000 was {{GrGl|Psi_V-shaped}} in [[Place:Boeotia|Boeotia]].<ref name=Heath-2003>{{cite book |last=Heath |first=Thomas L. |orig-year=1931 |year=2003 |title=A Manual of Greek Mathematics |edition={{small|[2003]}} reprint |publisher=Oxford University Press{{sup|[1931]}}; Dover Books{{sup|[2003]}} |location=Oxford, UK |pages=[https://books.google.com/books?id=_HZNr_mGFzQC&pg=PA14 14]&nbsp;ff |isbn=9780486154442 |url=https://books.google.com/books?id=_HZNr_mGFzQC&pg=PA14 |access-date=1 November 2013 |via=Google Books}}</ref>

The present system probably developed around Miletus in Ionia. 19th&nbsp;century classicists placed its development in the 3rd&nbsp;century&nbsp;BCE, the occasion of its first widespread use.<ref>{{cite book |last=Thompson |first=Edward M. |year=1893 |title=Handbook of Greek and Latin Palaeography |page=114 |place=New York, NY |publisher=D. Appleton}}</ref> More thorough [[Social:Modern archaeology|modern archaeology]] has caused the date to be pushed back at least to the 5th&nbsp;century&nbsp;BCE,<ref>{{cite book |department=The Packard Humanities Institute |publisher=[[Organization:Cornell University|Cornell University]] & [[Organization:Ohio State University|Ohio State University]] |title=Searchable Greek Inscriptions |article=IG I³ 1387 |id={{nobreak|IG I³ 1387}}&nbsp; ''also known as'' &nbsp;{{nobreak|IG I² 760}} |article-url=http://epigraphy.packhum.org/inscriptions/oi?ikey=1592&bookid=4&region=1 |access-date=1 November 2013}}</ref> a little before Athens abandoned its pre-Eucleidean alphabet in favour of Miletus's in 402&nbsp;BCE, and it may predate that by a century or two.<ref>{{cite book |last=Jeffery |first=Lilian H. |year=1961 |title=The Local Scripts of Archaic Greece |pages=38&nbsp;ff |place=Oxford, UK |publisher=Clarendon Press}}</ref> The present system uses the 24&nbsp;letters adopted under Eucleides, as well as three [[Phoenician alphabet|Phoenician]] and Ionic ones that had not been dropped from the Athenian alphabet (although kept for numbers): 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&nbsp;century&nbsp;BCE,<ref>{{cite book |department=The Packard Humanities Institute |publisher=[[Organization:Cornell University|Cornell University]] & [[Organization:Ohio State University|Ohio State University]] |title=Searchable Greek Inscriptions |article=Magnesia&nbsp;4 |id=Magnesia&nbsp;4&nbsp; ''also known as'' &nbsp;{{nobreak|Syll³ 695.b}} |article-url=http://epigraphy.packhum.org/inscriptions/oi?ikey=260445&bookid=509&region=8&subregion=27 |access-date=1 November 2013}}</ref> and its use is unattested in Athens until the 2nd&nbsp;century&nbsp;CE.<ref>{{cite book |department=The Packard Humanities Institute |publisher=[[Organization:Cornell University|Cornell University]] & [[Organization:Ohio State University|Ohio State University]] |title=Searchable Greek Inscriptions |article=IG II² 2776 |article-url=http://epigraphy.packhum.org/inscriptions/oi?ikey=5028&bookid=5&region=1 |access-date=1 November 2013}}</ref> (In general, Athenians resisted using the new numerals for the longest of any Greek state, but had fully adopted them by {{circa|50&nbsp;CE}}.<ref name=Heath-2003/>)

==Description<!--linked from 'Keraia'-->==
[[File:Greek minuscule numerals Cod.Const.Pal.Vet.f96r.svg|thumb|right|Greek numerals in a {{circa|lk=no|1100}} Byzantine manuscript of Hero of Alexandria's ''Metrika''. The first line contains the number "{{lang|grc|{{overline|͵θϡϟϛ}} δʹ ϛʹ}}", i.e. "{{nowrap|9,996 + {{frac|4}} + {{frac|6}}}}". It features each of the special numeral symbols sampi (ϡ), koppa (ϟ), and stigma (ϛ) in their minuscule forms.]]

Greek numerals are [[Decimal|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.<ref>{{cite web|url=http://gwydir.demon.co.uk/jo/numbers/greek/|title=Classical Greek Numbers|last1=Edkins|first1=Jo|year=2006|access-date=29 April 2013|archive-date=10 May 2013|archive-url=https://web.archive.org/web/20130510005614/http://gwydir.demon.co.uk/jo/numbers/greek/|url-status=dead}}</ref> (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.{{citation needed|date=June 2020}})

This [[Alphabetic numeral system|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 {{GrGl|Sigma classical}}{{GrGl|Mu classical}}{{GrGl|Alpha classical}}&nbsp;(200&nbsp;+&nbsp;40&nbsp;+&nbsp;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 [[Earth:Asia Minor|Asia Minor]] well into the Roman period.<ref name=heman>Heath, Thomas L. ''A Manual of Greek Mathematics'', [https://books.google.com/books?id=_HZNr_mGFzQC&pg=PA14 pp. 14 ff.] Oxford Univ. Press (Oxford), 1931. Reprinted Dover (Mineola), 2003. Accessed 1 November 2013.</ref>) In ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars: {{overline|α}}, {{overline|β}}, {{overline|γ}}, etc. In medieval manuscripts of the [[Religion:Book of Revelation|Book of Revelation]], the [[Unsolved:Number of the beast|number of the Beast]] 666 is written as {{overline|χξϛ}}&nbsp;(600&nbsp;+&nbsp;60&nbsp;+&nbsp;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]]s); for example {{nowrap|δʹ ϛʹ}} indicated {{nowrap|{{frac|4}} + {{frac|6}} {{=}} {{frac|5|12}}}}.

[[File:Add 19391 19-20.png|thumb|left|300px|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.]]
Although the Greek alphabet began with only majuscule forms, surviving [[Engineering:Papyrus|papyrus]] manuscripts from Egypt show that uncial and cursive [[Social:Greek minuscule|minuscule]] forms began early.{{clarification|date=August 2017|reason=When? Minuscule was introduced in Constantinople not earlier than the first half of the 9th century, see Talk Uppercase/Lowercase}} 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 ([[File:Greek Koppa cursive 02.svg|x16px]] and x16px) and simplified (x16px and x16px). 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 ({{GrGl|Digamma angular}}). By the Byzantine era, the letter was known as episemon and written as {{GrGl|Digamma cursive 02}} or {{GrGl|Digamma cursive 06}}. This eventually merged with the sigma-tau [[Social:Typographic ligature|ligature]] stigma ϛ ({{GrGl|Digamma cursive 07}} or {{GrGl|Digamma cursive 04}}).

{{anchor|keraia}}In modern Greek, a number of other changes have been made. Instead of extending an over bar over an entire number, the '''''keraia'''''<!--boldface per WP:R#PLA--> ({{lang|el|κεραία}}, <small>lit.</small>&nbsp;"hornlike projection") is marked to its upper right, a development of the short marks formerly used for single numbers and fractions. The modern ''keraia'' ({{keraia}}) is a symbol similar to the acute accent (´), the tonos (U+0384,΄) and the prime symbol (U+02B9, ʹ), but has its own [[Unicode]] character as U+0374. Alexander the Great's father Philip II of Macedon is thus known as {{lang|el|Φίλιππος Βʹ}} 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 ({{nowrap|2 × 1,000 + 10 + 9}}).

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.<ref>{{cite web|url=http://www.tlg.uci.edu/~opoudjis/unicode/numerals.html|title=Numerals: Stigma, Koppa, Sampi|author=Nick Nicholas|date=9 April 2005|access-date=2 March 2011|url-status=dead|archive-url=https://archive.today/20120805203248/http://www.tlg.uci.edu/~opoudjis/unicode/numerals.html|archive-date=2012-08-05}}</ref>

==Isopsephy==
{{main|Unsolved: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 ''[[Unsolved:Isopsephy|isopsephy]]''.  Similar practices for the Hebrew and English are called ''[[Unsolved:Gematria|gematria]]'' and [[Philosophy:English Qaballa|English Qaballa]], respectively.

==Table==
{{anchor|Chart|List}}
{| class="wikitable"
|-
! |Ancient!!Byzantine!!Modern!!Value !! !!Ancient!!Byzantine!!Modern!!Value !! !!Ancient!!Byzantine!!Modern!!Value
|-
|{{GrGl|Alpha classical}}||α̅||{{lang|el|Αʹ}}||1
|rowspan=19|
|{{GrGl|Iota classical}}||ι̅||{{lang|el|Ιʹ}}||10
|rowspan=19|
|{{GrGl|Rho classical}}||ρ̅||{{lang|el|Ρʹ}}||[[100 (number)|100]]
|-
|{{GrGl|Beta classical}}||β̅||{{lang|el|Βʹ}}||2||{{GrGl|Kappa classical}}||κ̅||{{lang|el|Κʹ}}||[[20 (number)|20]]||{{GrGl|Sigma classical}}||σ̅||{{lang|el|Σʹ}}||[[200 (number)|200]]
|-
|{{GrGl|Gamma classical}}||γ̅||{{lang|el|Γʹ}}||3||{{GrGl|Lambda classical}}||λ̅||{{lang|el|Λʹ}}||[[30 (number)|30]]||{{GrGl|Tau classical}}||τ̅||Τʹ||[[300 (number)|300]]
|-
|{{GrGl|Delta classical}}||δ̅||{{lang|el|Δʹ}}||4||{{GrGl|Mu classical}}||μ̅||{{lang|el|Μʹ}}||[[40 (number)|40]]||{{GrGl|Upsilon classical}}||υ̅||{{lang|el|Υʹ}}||[[400 (number)|400]]
|-
|{{GrGl|Epsilon classical}}||ε̅||{{lang|el|Εʹ}}||5||{{GrGl|Nu classical}}||ν̅||{{lang|el|Νʹ}}||[[50 (number)|50]]||{{GrGl|Phi classical}}||φ̅||{{lang|el|Φʹ}}||[[500 (number)|500]]
|-
|{{GrGl|Digamma oblique}}<br>{{GrGl|Digamma angular}}||{{GrGl|Digamma cursive 02}}&nbsp;and&nbsp;{{GrGl|Digamma cursive 04}}<br>{{GrGl|Digamma cursive 06}}&nbsp;and&nbsp;{{GrGl|Digamma cursive 07}}||{{lang|el|Ϛʹ}}<br>Ϝʹ<br>ΣΤʹ||6||{{GrGl|Xi classical}}||ξ̅||Ξʹ||[[60 (number)|60]]||{{GrGl|Chi classical}}||χ̅||{{lang|el|Χʹ}}||[[600 (number)|600]]
|-
|{{GrGl|Zeta classical}}||ζ̅||{{lang|el|Ζʹ}}||7||{{GrGl|Omicron classical}}||ο̅||{{lang|el|Οʹ}}||[[70 (number)|70]]||{{GrGl|Psi classical}}||ψ̅||{{lang|el|Ψʹ}}||[[700 (number)|700]]
|-
|{{GrGl|Eta classical}}||η̅||{{lang|el|Ηʹ}}||8||{{GrGl|Pi classical}}||π̅||{{lang|el|Πʹ}}||[[80 (number)|80]]||{{GrGl|Omega classical}}||ω̅||{{lang|el|Ωʹ}}||[[800 (number)|800]]
|-
|{{GrGl|Theta classical}}||θ̅||{{lang|el|Θʹ}}||9||{{GrGl|Koppa normal}}<br>{{GrGl|Koppa cursive 01}}||{{GrGl|Koppa cursive 02}}&nbsp;and&nbsp;{{GrGl|Koppa cursive 04}}<br>{{GrGl|Koppa cursive 03}}&nbsp;and&nbsp;{{GrGl|Koppa cursive 05}}||Ϟʹ<br>Ϙʹ||[[90 (number)|90]]||{{GrGl|Sampi Ionian}}<br>{{GrGl|Sampi palaeographic 05}}&nbsp;and&nbsp;{{GrGl|Sampi palaeographic 15}}<br>{{GrGl|Sampi palaeographic 06}}&nbsp;and&nbsp;{{GrGl|Sampi palaeographic 09}}||{{GrGl|Sampi palaeographic 03}}&nbsp;and&nbsp;{{GrGl|Sampi palaeographic 07}}<br>{{GrGl|Sampi palaeographic 08}}<br>{{GrGl|Sampi palaeographic 10}}&nbsp;and&nbsp;{{GrGl|Sampi palaeographic 11}}<br>{{GrGl|Sampi palaeographic 14}}&nbsp;and&nbsp;{{GrGl|Sampi palaeographic 13}}<br>[[File:Sampi.svg|x16px]]||Ϡʹ<br>Ͳʹ||[[900 (number)|900]]
|-
! |!!!!!! !! !!!!!! !! !!!!!!
|-
||{{GrGl|Sampi 1000}}&nbsp;and&nbsp;{{GrGl|Sampi 1000 (2)}}||͵α||,Α||[[1000 (number)|1000]]||<sup>[[File:Greek_Iota_classical.svg|x8px]]</sup><sub>x16px</sub>||͵ι||,Ι||10000||<sup>x8px</sup><sub>x16px</sub>||͵ρ||,Ρ||100000
|-
|<sup>[[File:Greek_Beta_classical.svg|x8px]]</sup><sub>x16px</sub>||͵β||,Β||[[2000 (number)|2000]]||<sup>x8px</sup><sub>x16px</sub>||͵κ||,Κ||20000||<sup>x8px</sup><sub>x16px</sub>||͵σ||,Σ||200000
|-
|<sup>[[File:Greek_Gamma_classical.svg|x8px]]</sup><sub>x16px</sub>||͵{{GrGl|Gamma 02}}||,Γ||[[3000 (number)|3000]]||<sup>x8px</sup><sub>x16px</sub>||͵λ||,Λ||30000||<sup>x8px</sup><sub>x16px</sub>||͵τ||,Τ||300000
|-
|<sup>[[File:Greek_Delta_classical.svg|x8px]]</sup><sub>x16px</sub>||͵{{GrGl|Delta classical}}||,Δ||[[4000 (number)|4000]]||<sup>x8px</sup><sub>x16px</sub>||͵μ||,Μ||40000||<sup>x8px</sup><sub>x16px</sub>||͵υ||,Υ||400000
|-
|<sup>[[File:Greek_Epsilon_classical.svg|x8px]]</sup><sub>x16px</sub>||͵ε||,Ε||[[5000 (number)|5000]]||<sup>x8px</sup><sub>x16px</sub>||͵ν||,Ν||50000||<sup>x8px</sup><sub>x16px</sub>||͵φ||,Φ||500000
|-
|<sup>[[File:Greek_Digamma_oblique.svg|x8px]]</sup><sub>x16px</sub><br><sup>x8px</sup><sub>x16px</sub>||͵{{GrGl|Digamma cursive 02}}&nbsp;and&nbsp;͵{{GrGl|Digamma cursive 04}}<br>͵{{GrGl|Digamma cursive 06}}&nbsp;and&nbsp;͵{{GrGl|Digamma cursive 07}}||,Ϛ<br>,Ϝ<br>,ΣΤ||[[6000 (number)|6000]]||<sup>x8px</sup><sub>x16px</sub>||͵ξ||,Ξ||60000||<sup>x8px</sup><sub>x16px</sub>||͵χ||,Χ||600000
|-
|<sup>[[File:Greek_Zeta_classical.svg|x8px]]</sup><sub>x16px</sub>||͵ζ||,Ζ||[[7000 (number)|7000]]||<sup>x8px</sup><sub>x16px</sub>||͵ο||,Ο||70000||<sup>x8px</sup><sub>x16px</sub>||͵ψ||,Ψ||700000
|-
|<sup>[[File:Greek_Eta_classical.svg|x8px]]</sup><sub>x16px</sub>||͵η||,Η||[[8000 (number)|8000]]||<sup>x8px</sup><sub>x16px</sub>||͵π||,Π||80000||<sup>x8px</sup><sub>x16px</sub>||͵ω||,Ω||800000
|-
|{{GrGl|Sampi 9000}}||͵θ||,Θ||[[9000 (number)|9000]]||<sup>[[File:Greek_Koppa_normal.svg|x8px]]</sup><sub>x16px</sub><br><sup>x8px</sup><sub>x16px</sub>||͵{{GrGl|Koppa cursive 02}}&nbsp;and&nbsp;͵{{GrGl|Koppa cursive 04}}<br>͵{{GrGl|Koppa cursive 03}}&nbsp;and&nbsp;͵{{GrGl|Koppa cursive 05}}||,Ϟ<br>,Ϙ||90000||<sup>x8px</sup><sub>x16px</sub><br><sup>x8px</sup><sub>x16px</sub>&nbsp;and&nbsp;<sup>x8px</sup><sub>x16px</sub><br><sup>x8px</sup><sub>x16px</sub>&nbsp;and&nbsp;<sup>x8px</sup><sub>x16px</sub>||͵{{GrGl|Sampi palaeographic 03}}&nbsp;and&nbsp;͵{{GrGl|Sampi palaeographic 07}}<br>͵{{GrGl|Sampi palaeographic 08}}<br>͵{{GrGl|Sampi palaeographic 10}}&nbsp;and&nbsp;͵{{GrGl|Sampi palaeographic 11}}<br>͵{{GrGl|Sampi palaeographic 14}}&nbsp;and&nbsp;͵{{GrGl|Sampi palaeographic 13}}<br>͵[[File:Sampi.svg|x16px]]||,Ϡ<br>,Ͳ||900000
|}
* Alternatively, sub-sections of manuscripts are sometimes numbered by lowercase characters (αʹ. βʹ. γʹ. δʹ. εʹ. ϛʹ. ζʹ. ηʹ. θʹ.).
* In Ancient Greek, [[Myriad|myriad]] notation is used for multiples of 10,000, for example {{overset|β|Μ}} for 20,000 or {{overset|ρκγ|Μ}}{{overline|͵δφξζ}} (also written on the line as {{overline|ρκγ}}Μ&nbsp;{{overline|͵δφξζ}}) for 1,234,567.<ref name=mactutor>[http://www-history.mcs.st-andrews.ac.uk/PrintHT/Greek_numbers.html Greek number systems - MacTutor]</ref>

==Higher numbers==
In his text ''[[Astronomy:The Sand Reckoner|The Sand Reckoner]]'', the natural philosopher [[Biography:Archimedes|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 [[Astronomy:The Sand Reckoner#Naming large numbers|new numeral scheme]] with much greater range.

[[Biography:Pappus of Alexandria|Pappus of Alexandria]] reports that [[Biography:Apollonius of Perga|Apollonius of Perga]] developed a simpler system based on [[Exponentiation|powers]] of the myriad; {{overset|α|Μ}} was 10,000, {{overset|β|Μ}} was 10,000<sup>2</sup> = 100,000,000, {{overset|γ|Μ}} was 10,000<sup>3</sup> = 10<sup>12</sup> and so on.<ref name=mactutor/>

==Zero==
[[File:P. Lund, Inv. 35a.jpg|thumb|right|300px|Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus]]
{{anchor|Hellenistic zero}}<!--linked from "Number"-->
Hellenistic [[Astronomy:Astronomer|astronomer]]s extended alphabetic Greek numerals into a [[Sexagesimal|sexagesimal]] [[Positional notation|positional]] [[Numeral system|numbering system]] by limiting each position to a maximum value of 50&nbsp;+&nbsp;9 and including a special symbol for zero, 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 [[Biography:Hipparchus|Hipparchus]] {{circa|140 BC}}. It was then used by [[Biography:Ptolemy|Ptolemy]] ({{circa|140}}), [[Biography:Theon of Alexandria|Theon]] ({{circa|380}}) and Theon's daughter Hypatia (died&nbsp;415). The symbol for zero is clearly different from that of the value for 70, omicron or "&omicron;". 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 first fairly extensive trigonometric table, there were 360&nbsp;rows, portions of which looked as follows:

: <math>
\begin{array}{ccc} \pi\varepsilon\varrho\iota\varphi\varepsilon\varrho\varepsilon\iota\tilde\omega\nu & \varepsilon\overset{\text{'}}\upsilon\vartheta\varepsilon\iota\tilde\omega\nu & \overset{\text{`}}\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\
\begin{array}{|l|} \hline \pi\delta\angle' \\  \pi\varepsilon \\  \pi\varepsilon\angle' \\  \hline  \pi\stigma \\  \pi\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\stigma & \vartheta \\  \hline \end{array} & \begin{array}{|r|r|r|r|} \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{array}
\end{array}
</math>
Each number in the first column, labeled {{math|{{lang|grc|περιφερειῶν}},}} ["regions"] is the number of degrees of arc on a circle.  Each number in the second column, labeled {{math|{{lang|grc|εὐθειῶν}},}} ["straight lines" or "segments"] is the length of the corresponding chord of the circle, when the diameter is 120.  Thus {{math|πδ}} represents an 84° arc, and the ∠′ after it means one-half, so that πδ∠′ means {{frac|84|1|2}}°.  In the next column we see {{math|π&nbsp;μα&nbsp;γ}}&nbsp;, meaning {{nowrap|&nbsp; 80 + {{sfrac|41|60}} + {{sfrac|3|60²}}}}.  That is the length of the chord corresponding to an arc of {{frac|84|1|2}}° when the diameter of the circle is 120. The next column, labeled {{math|{{lang|grc|ἐξηκοστῶν}},}} 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|linear interpolation]].

The Greek [[Sexagesimal|sexagesimal]] placeholder or zero symbol changed over time: The symbol used on [[Engineering:Papyrus|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 [[Place:Byzantine Empire|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 {{math|οὐδέν}} meaning "nothing".<ref>
{{cite book
 | orig-year = 1957 | year = 1969
 | title = The Exact Sciences in Antiquity
 | edition = 2, reprint    | pages = 13–14, plate&nbsp;2
 | publisher = [[Social:Dover Publications|Dover Publications]]
 | isbn = 978-0-486-22332-2
}}
</ref><ref>
{{cite web
 |first=Raymond |last=Mercier
 |title=Consideration of the Greek symbol 'zero'
 |url= http://www.raymondm.co.uk/prog/GreekZeroSign.pdf
}} — gives numerous examples
</ref> 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]] 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 [[Astronomy:Moon|Moon]] and either the center of the [[Astronomy:Sun|Sun]] (for [[Earth:Solar eclipse|solar eclipse]]s) or the center of [[Astronomy:Earth|Earth]]'s shadow (for [[Astronomy:Lunar eclipse|lunar eclipse]]s). All of these zeros took the form {{nowrap|ο {{!}} ο ο}}, 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.<ref>
{{cite book
 |orig-year=100–170&nbsp;CE |year=1998
 |chapter=Book&nbsp;VI
 |pages=306–307
 |title=Ptolemy's [[Astronomy:Almagest|Almagest]]
 |translator-last=Toomer |translator-first=G.J.
 |translator-link=G. J. Toomer
 |place=Princeton, NJ
 |publisher=Princeton University Press
}}
</ref>
{{charmap
| 1018A | name1 = GREEK ZERO SIGN
}}

==See also==

* {{annotated link|Alphabetic numeral system}}
* {{annotated link|Attic numerals}}
* {{annotated link|Cyrillic numerals}}
* {{annotated link|Hebrew numerals}}, based on the Greek system
* {{annotated link|History of ancient numeral systems}}
* {{annotated link|History of arithmetic}}
* {{annotated link|Social:History of communication}}
* {{annotated link|Unsolved:Isopsephy}}
* {{annotated link|List of numeral system topics}}
* {{annotated link|List of numeral systems}}
* {{annotated link|Unsolved:Number of the beast}}
* {{annotated link|Roman numerals}}

==References==
{{Reflist|30em}}

==External links==
* [http://www.russellcottrell.com/greek/utilities/GreekNumberConverter.htm The Greek Number Converter]

{{Ancient Greek mathematics}}


{{DEFAULTSORT:Greek Numerals}}
[[Category:Numeral systems]]
[[Category:Numerals]]

{{Sourceattribution|Greek numerals}}