Editing Greek numerals (section)

==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 Empire|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 (letter)|koppa]] (ϟ), and [[stigma (letter)|stigma]] (ϛ) in their [[Greek minuscule|minuscule]] forms.]]

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 (letter)|alpha]] to [[theta (letter)|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 (letter)|iota]] to [[koppa (letter)|koppa]]. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well, from [[rho (letter)|rho]] to [[sampi (letter)|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 [[Asia Minor]] well into the [[Roman Greece|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, New York|Mineola]]), 2003. Accessed 1 November 2013.</ref>) In ancient and medieval manuscripts, these numerals were eventually distinguished from letters using [[overbar]]s: {{overline|α}}, {{overline|β}}, {{overline|γ}}, etc. In medieval manuscripts of the [[Book of Revelation]], the [[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 [[Claudius Ptolemy|Ptolemy]]'s [[Ptolemy's Geography|''Geography'']], using Greek numerals for its [[Geographic coordinate system|graticule]]: 52–63°N of the [[equator]] and 6–33°E from Ptolemy's [[Prime Meridian]] at the [[Fortunate Isles]].]]
Although the [[Greek alphabet#Letter shapes|Greek alphabet]] began with only [[majuscule]] forms, surviving [[papyrus]] manuscripts from [[Roman Egypt|Egypt]] show that [[uncial]] and [[cursive]] [[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 [[File:Greek Koppa cursive 03.svg|x16px]]) and simplified ([[File:Greek Koppa cursive 04.svg|x16px]] and [[File:Greek Koppa cursive 05.svg|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 Greek|Byzantine era]], the letter was known as [[Digamma#Episemon|episemon]] and written as {{GrGl|Digamma cursive 02}} or {{GrGl|Digamma cursive 06}}. This eventually merged with the [[sigma (letter)|sigma]]-[[tau (letter)|tau]] [[Typographic ligature|ligature]] [[stigma (letter)|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>