Editing Greek numerals (section)

==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 civilization|Hellenistic]] [[astronomer]]s extended alphabetic Greek numerals into a [[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 [[0 (number)|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 [[Hipparchus]] {{circa|140 BC}}. It was then used by [[Ptolemy]] ({{circa|140 BC|lk=no}}), [[Theon of Alexandria|Theon]] ({{circa|380  AD|lk=no}}) and Theon's daughter [[Hypatia of Alexandria|Hypatia]] ({{died-in|415 AD}}). The symbol for zero is clearly different from that of the value for 70, [[omicron]] or "[[omicron|&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 {{math|∠′}} after it means one-half, so that {{math|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|ἑξηκοστῶν}},}} 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]].

The Greek [[sexagesimal]] placeholder or zero symbol changed over time: The symbol used on [[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, later the overbar was omitted in [[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
 | last = Neugebauer | first = Otto | author-link = Otto E. Neugebauer
 | orig-year = 1957 | year = 1969
 | title = The Exact Sciences in Antiquity
 | edition = 2, reprint    | pages = 13–14, plate&nbsp;2
 | publisher = [[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]] number, and zero was usually omitted when it was the integer.

Some of Ptolemy's true zeros appeared in the first line of each of his eclipse tables, where they were a measure of the angular separation between the center of the [[Moon]] and either the center of the [[Sun]] (for [[solar eclipse]]s) or the center of [[Earth]]'s shadow (for [[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
 |first=Claudius |last=Ptolemy |author-link=Ptolemy
 |orig-year=100–170&nbsp;AD |year=1998
 |chapter=Book&nbsp;VI
 |pages=306–307
 |title=Ptolemy's [[Almagest]]
 |translator-last=Toomer |translator-first=G.J.
 |translator-link=G. J. Toomer
 |place=Princeton, NJ
 |publisher=Princeton University Press
}}
</ref>

The Greek zero was added to Unicode in Version 4.1.0 at {{unichar|1018A}}.<ref>{{Cite Unicode|4.1.0}}</ref>