Greek numerals
Template:Short description Template:Hatnote Template:Lead too short Template:Numeral systems Template:Contains special characters Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, is a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those in which Roman numerals are still used in the Western world. For ordinary cardinal numbers, however, modern Greece uses Arabic numerals.
History
The Minoan and Mycenaean civilizations' Linear A and Linear B alphabets used a different system, called Aegean numerals, which included number-only symbols for powers of ten: Template:Lang = 1, Template:Lang = 10, Template:Lang = 100, Template:Lang = 1,000, and Template:Lang = 10,000.<ref name=Verdun-2007-03-20>Template:Cite web</ref>
Attic numerals composed another system that came into use perhaps in the 7th century BC. They were acrophonic, derived (after the initial one) from the first letters of the names of the numbers represented. They ran Template:GrGl = 1, Template:GrGl = 5, Template:GrGl = 10, Template:GrGl = 100, Template:GrGl = 1,000, and Template:GrGl = 10,000. The numbers 50, 500, 5,000, and 50,000 were represented by the letter Template:GrGl with minuscule powers of ten written in the top-right corner: 
, 
, 
, and 
.<ref name=Verdun-2007-03-20/> One-half was represented by Template:Script (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 Template:GrGl in Boeotia.<ref name=Heath-2003>Template:Cite book</ref>
The present system probably developed around Miletus in Ionia. 19th century classicists placed its development in the 3rd century BC, the occasion of its first widespread use.<ref>Template:Cite book</ref> More thorough modern archaeology has caused the date to be pushed back at least to the 5th century BC,<ref>Template:Cite book</ref> a little before Athens abandoned its pre-Eucleidean alphabet in favour of Miletus's in 402 BC, and it may predate that by a century or two.<ref>Template:Cite book</ref> The present system uses the 24 letters adopted under Eucleides, as well as three 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 century BC,<ref>Template:Cite book</ref> and its use is unattested in Athens until the 2nd century CE.<ref>Template:Cite book</ref> (In general, Athenians resisted using the new numerals for the longest of any Greek state, but had fully adopted them by Template:Circa.<ref name=Heath-2003/>)
Description
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 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>Template:Cite web</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.Template:Citation needed)
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 Template:GrGlTemplate:GrGlTemplate:GrGl (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 well into the Roman period.<ref name=heman>Heath, Thomas L. A Manual of Greek Mathematics, 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: Template:Overline, Template:Overline, Template:Overline, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as Template:Overline (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 (Template:Keraia); γTemplate:Keraia indicated one third, δTemplate:Keraia one fourth and so on. As an exception, special symbol ∠Template:Keraia indicated one half, and γ°Template:Keraia or γoTemplate:Keraia was two-thirds. These fractions were additive (also known as Egyptian fractions); for example Template:Nowrap indicated Template:Nowrap.
Although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early.Template:Clarification 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 (Template:GrGl). By the Byzantine era, the letter was known as episemon and written as Template:GrGl or Template:GrGl. This eventually merged with the sigma-tau ligature stigma ϛ (Template:GrGl or Template:GrGl).
Template:AnchorIn modern Greek, a number of other changes have been made. Instead of extending an over bar over an entire number, the keraia (Template:Lang, 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 (Template: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 Template:Lang in modern Greek. A lower left keraia (Unicode: U+0375, "Greek Lower Numeral Sign") is now standard for distinguishing thousands: 2019 is represented as Template:LkeraiaΒΙΘTemplate:Keraia (Template:Nowrap).
The declining use of ligatures in the 20th century also means that stigma is frequently written as the separate letters ΣΤTemplate:Keraia, although a single keraia is used for the group.<ref>Template:Cite web</ref>
Isopsephy
Template:Main 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. Similar practices for the Hebrew and English are called gematria and English Qaballa, respectively.
Table
- Sub-sections of Greek manuscripts are sometimes numbered by lowercase characters (αTemplate:Keraia. βTemplate:Keraia. γTemplate:Keraia. δTemplate:Keraia. εTemplate:Keraia. ϛTemplate:Keraia. ζTemplate:Keraia. ηTemplate:Keraia. θTemplate:Keraia.).
- In Ancient Greek, myriad notation is used for multiples of 10,000, for example Template:Overset for 20,000 or Template:OversetTemplate:Overline for 1,234,567 (also written on the line as Template:OverlineΜ Template:Overline).<ref name=mactutor>Template:Cite web</ref>
Higher numbers
In his text The Sand Reckoner, Archimedes gives an upper bound of the number of grains of sand required to fill the entire universe, using an estimate of its size current in his time. His essay demonstrated a easily visualized contradiction of the then-held adage that it was impossible to name a quantity "greater than the number of grains of sand on a beach", or in the entire world. In order to do that, he devised a new enumeration scheme with much greater range than any of those shown above.
Pappus of Alexandria reports that Apollonius of Perga developed a simpler system based on powers of the myriad:
- Template:Overset was 10,000 = Template:10^
- Template:Overset was (10,000)2 = 100,000,000 = Template:10^
- Template:Overset was (10,000)3 = Template:10^
and so on.<ref name=mactutor/>
Zero
Template:Anchor Hellenistic astronomers extended alphabetic Greek numerals into a sexagesimal positional numbering system by limiting each position to a maximum value of Template:Nobr which uses only the letters up through nu (Template:Math) and included otherwise unused omicron (Template:Math) as a special symbol for zero. Omicron as zero 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 Template:Circa. It was then used by Ptolemy (Template:Circa), Theon (Template:Circa) and Theon's daughter Hypatia (Template:Died-in). The symbol omicron or [[omicron|'Template:Math']] as used for zero in astronomical and mathematical data tables is clearly different from its conventional use as the value for 70. 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 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 Template:Math ("regions") is the number of degrees of arc on a circle. Each number in the second column, labeled Template:Math ("straight lines" or "segments") is the length of the corresponding chord of the circle, when the diameter is 120 units. Thus Template:Math represents an 84° arc, and the Template:Math after it means one-half, so that Template:Math means Template:Sfrac°. In the next column we see Template:Nobr meaning Template:Nowrap. That is the length of the chord corresponding to an arc of Template:Sfrac° when the diameter of the circle is 120 units. The next column, labeled Template:Math 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 papyri during the second century was a very small circle with an overbar several diameters long (Template:Math), terminated or not at both ends in various ways. Later, the overbar shortened to only one letter-diameter, similar to the modern 'o'+macron (ō). It was still being used in late medieval Arabic manuscripts whenever alphabetic numerals were used. In later Byzantine manuscripts even the minimal overbar was omitted, leaving a bare 'ο' (omicron).<ref> Template:Cite book </ref><ref> Template:Cite web — gives numerous examples </ref> This gradual change from an invented symbol Template:Math to 'ο' does not support the hypothesis that the omicron was the initial of Template:Math 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 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 eclipses) or the center of Earth's shadow (for lunar eclipses). All of these zeros took the form Template:Nowrap, 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> Template:Cite book Originally published under the (translated) title Mathematical Syntaxis. </ref>
The Greek zero was added to Unicode in Version 4.1.0 at Template:Unichar.<ref>Template:Cite Unicode</ref>
See also
- Template:Annotated link
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- Template:Annotated link
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- Template:Annotated link (acrophonic, not alphabetic, numerals)
- Template:Annotated link, based on the Greek system
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References
External links
Template:Ancient Greek mathematics
Template:Greek language Template:Ancient Greece topics Template:Use dmy dates