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词条 Greek numerals
释义

  1. History

  2. Description

  3. Isopsephy (Gematria)

  4. Table

  5. Higher numbers

  6. Zero

  7. See also

  8. References

  9. External links

{{hatnote|The numerical signs ʹ and ͵ redirect here. For the accent mark ´, see Acute accent.}}{{numeral systems}}{{Special characters}}

Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are 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 elsewhere in the West. For ordinary cardinal numbers, however, 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 specialised symbols for numbers: {{lang|gmy|𐄇}} = 1, {{lang|gmy|𐄐}} = 10, {{lang|gmy|𐄙}} = 100, {{lang|gmy|𐄢}} = 1000, and {{lang|gmy|𐄫}} = 10000.[1]

Attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived (after the initial one) from the first letters of the names of the numbers represented. They ran {{GrGl|Zeta archaic}} = 1, {{GrGl|Pi archaic}} = 5, {{GrGl|Delta 04}} = 10, {{GrGl|Eta classical}} = 100, {{GrGl|Chi normal}} = 1,000, and {{GrGl|Mu classical}} = 10,000. {{nowrap|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: , , , and .[1] The same system was used outside of Attica, but the symbols varied with the local alphabets: in Boeotia, {{GrGl|Psi_V-shaped}} was 1,000.[2]

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.[3] More thorough modern archaeology has caused the date to be pushed back at least to the 5th century BC,[4] a little before Athens abandoned its pre-Euclidean alphabet in favour of Miletus's in 402 BC, and it may predate that by a century or two.[5] The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over: 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[6] and its use is unattested in Athens until the 2nd century AD.[7] (In general, Athens resisted the use of the new numerals for the longest of any of the Greek states but had fully adopted them by {{circa|lk=no|AD 50}}.[2])

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.[8] (The fact that this was not the traditional location of sampi or its possible predecessor san has led classicists to conclude that it was no longer in use even locally 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 {{GrGl|Sigma classical}}{{GrGl|Mu classical}}{{GrGl|Alpha classical}} (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.[2]) 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 Book of Revelation, the number of the Beast 666 is written as {{overline|χξϛ}} (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. These fractions were additive (also known as Egyptian fractions); for example {{nowrap|δʹ ϛʹ}} indicated {{nowrap|{{frac|4}} + {{frac|6}} {{=}} {{frac|5|12}}}}.

Although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive 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 ( and ) and simplified ( and ). The numeral for 6 changed several times. During antiquity, the original letter form of digamma ({{GrGl|Digamma oblique}}) 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 ligature stigma ({{GrGl|Digamma cursive 07}} or {{GrGl|Digamma cursive 04}}).

In modern Greek, a number of other changes have been made. Instead of extending an over bar over an entire number, the keraia ({{lang|el|κεραία}}, 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 (ʹ) 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: 2,018 is represented as ͵ΒΙΗʹ ({{nowrap|2 × 1,000 + 10 + 8}}).

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.[9]

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 (gematria).

Table

{{anchor|Chart|List}}
AncientByzantineModernValue AncientByzantineModernValue AncientByzantineModernValue AncientByzantineModernValue
Alpha classical}}α}}el|Αʹ}}1Iota classical}}ι}}el|Ιʹ}}10{{GrGl|Rho classical}}ρ}}el|Ρʹ}}100{{GrGl|Sampi 1000}} & {{GrGl|Sampi 1000 (2)}}͵αel|͵Α}}1000
Beta classical}}β}}el|Βʹ}}2Kappa classical}}κ}}el|Κʹ}}20Sigma classical}}σ}}el|Σʹ}}200͵βel|͵Β}}2000
Gamma classical}}γ}}el|Γʹ}}3Lambda classical}}λ}}el|Λʹ}}30Tau classical}}τ}}Τʹ300Gamma 02}}el|͵Γ}}3000
Delta classical}}δ}}el|Δʹ}}4Mu classical}}μ}}el|Μʹ}}40Upsilon classical}}υ}}el|Υʹ}}400Delta classical}}el|͵Δ}}4000
Epsilon classical}}ε}}el|Εʹ}}5Nu classical}}ν}}el|Νʹ}}50Phi classical}}φ}}el|Φʹ}}500͵εel|͵Ε}}5000
Digamma oblique}}
{{GrGl|Digamma angular}}		Digamma cursive 02}} & {{GrGl|Digamma cursive 04}}
{{GrGl|Digamma cursive 06}} & {{GrGl|Digamma cursive 07}}		el|Ϛʹ}}
{{lang|el|ΣΤʹ}}		6Xi classical}}ξ}}Ξʹ60Chi classical}}χ}}el|Χʹ}}600Digamma cursive 02}} & ͵{{GrGl|Digamma cursive 04}}
͵{{GrGl|Digamma cursive 06}} & ͵{{GrGl|Digamma cursive 07}}		el|͵Ϛ}}6000
Zeta classical}}ζ}}el|Ζʹ}}7Omicron classical}}ο}}el|Οʹ}}70Psi classical}}ψ}}el|Ψʹ}}700͵ζel|͵Z}}7000
Eta classical}}η}}el|Ηʹ}}8Pi classical}}π}}el|Πʹ}}80Omega classical}}ω}}el|Ωʹ}}800͵ηel|͵H}}8000
Theta classical}}θ}}el|Θʹ}}9Koppa normal}}
{{GrGl|Koppa cursive 01}}		Koppa cursive 02}} & {{GrGl|Koppa cursive 04}}
{{GrGl|Koppa cursive 03}} & {{GrGl|Koppa cursive 05}}		el|Ϟʹ}}90Sampi Ionian}}
{{GrGl|Sampi palaeographic 05}} & {{GrGl|Sampi palaeographic 15}}
{{GrGl|Sampi palaeographic 06}} & {{GrGl|Sampi palaeographic 09}}		Sampi palaeographic 03}} & {{GrGl|Sampi palaeographic 07}}
{{GrGl|Sampi palaeographic 08}}
{{GrGl|Sampi palaeographic 10}} & {{GrGl|Sampi palaeographic 11}}
{{GrGl|Sampi palaeographic 14}} & {{GrGl|Sampi palaeographic 13}}
el|Ϡʹ}}900Sampi 9000}}͵θel|͵Θ}}9000
  • Alternatively, sub-sections of manuscripts are sometimes numbered by lowercase characters {αʹ. βʹ. γʹ. δʹ. εʹ. ϛʹ. ζʹ. ηʹ. θʹ.}
  • In Ancient Greek, 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 {{overline|͵δφξζ}}) for 1,234,567.[10]

Higher numbers

In his text The Sand Reckoner, the natural philosopher 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 new numeral scheme with much greater range.

Pappus of Alexandria reports that Apollonius of Perga developed a simpler system based on powers of the myriad; {{overset|α|Μ}} was 10,000, {{overset|β|Μ}} was 10,0002 = 100,000,000, {{overset|γ|Μ}} was 10,0003 = 1012 and so on.[10]

Zero

{{anchor|Hellenistic zero}}

Hellenistic astronomers extended alphabetic Greek numerals into a sexagesimal positional numbering system by limiting each position to a maximum value of 50 + 9 and including a special symbol for zero, which was also used alone like today's modern zero, more than as a simple placeholder. However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.) — they were not used for the integral part of a number. This system was probably adapted from Babylonian numerals by Hipparchus {{circa|140 BC}}. It was then used by Ptolemy ({{circa|140}}), Theon ({{circa|380}}) and Theon's daughter Hypatia (died 415).

In Ptolemy's table of chords, the first fairly extensive trigonometric table, there were 360 rows, portions of which looked as follows:

Each number in the first column, labeled {{lang|grc|περιφερειῶν}}, is the number of degrees of arc on a circle. Each number in the second column, labeled {{lang|grc|εὐθειῶν}}, is the length of the corresponding chord of the circle, when the diameter is 120. Thus 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 π μα γ, meaning 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 {{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.

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, 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 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 οὐδέν meaning "nothing".[11][12] 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 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 {{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.[13]

See also

  • Attic numerals
  • Cyrillic numerals
  • Gematria
  • Greek numerals in Unicode (acrophonic, not alphabetic, numerals)
  • Isopsephy
  • Number of the Beast

References

1. ^1 {{cite web|url=http://www.dma.ens.fr/culturemath/histoire%20des%20maths/htm/Verdan/Verdan.htm|title=Systèmes numéraux en Grèce ancienne: description et mise en perspective historique|author=Samuel Verdan|date=20 Mar 2007 |accessdate=2 Mar 2011|language=fr}}
2. ^1 2 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.
3. ^Thompson, Edward M. Handbook of Greek and Latin Palaeography, p. 114. D. Appleton (New York), 1893.
4. ^The Packard Humanities Institute (Cornell & Ohio State Universities). Searchable Greek Inscriptions: "IG I3 1387" [also known as IG I2 760]. Accessed 1 November 2013.
5. ^Jeffery, Lilian H. The Local Scripts of Archaic Greece, pp. 38 ff. Clarendon (Oxford), 1961.
6. ^The Packard Humanities Institute (Cornell & Ohio State Universities). Searchable Greek Inscriptions: "Magnesia 4" [also known as Syll³ 695.b]. Accessed 1 November 2013.
7. ^The Packard Humanities Institute (Cornell & Ohio State Universities). Searchable Greek Inscriptions: "IG II² 2776". Accessed 1 November 2013.
8. ^{{cite web|url=http://gwydir.demon.co.uk/jo/numbers/greek/|title=Classical Greek Numbers|last1=Edkins|first1=Jo|year=2006|accessdate=29 Apr 2013}}
9. ^{{cite web|url=http://www.tlg.uci.edu/~opoudjis/unicode/numerals.html|title=Numerals: Stigma, Koppa, Sampi|author=Nick Nicholas|date=9 Apr 2005|accessdate=2 Mar 2011|deadurl=yes|archiveurl=https://archive.is/20120805203248/http://www.tlg.uci.edu/~opoudjis/unicode/numerals.html|archivedate=2012-08-05|df=}}
10. ^1 Greek number systems - MacTutor
11. ^{{Cite book | edition = 2 | publisher = Dover Publications | last = Neugebauer | first = Otto | author-link = Otto E. Neugebauer | title = The Exact Sciences in Antiquity | origyear = 1957 | year = 1969 | isbn = 978-0-486-22332-2 | pages = 13–14, plate 2}}
12. ^{{cite web|first=Raymond |last=Mercier|url= http://www.raymondm.co.uk/prog/GreekZeroSign.pdf |title=Consideration of the Greek symbol 'zero' }} Numerous examples
13. ^Ptolemy's Almagest, translated by G. J. Toomer, Book VI, (Princeton, NJ: Princeton University Press, 1998), pp. 306–7.

External links

{{Commons category|Greek numerals}}
  • The Greek Number Converter
{{Greek language}}{{Ancient Greece topics}}{{DEFAULTSORT:Greek Numerals}}

3 : Numeral systems|Numerals|Greek mathematics
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