Greek numerals

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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.

Contents

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: 𐄇 = 1, 𐄐 = 10, 𐄙 = 100, 𐄢 = 1000, and 𐄫 = 10000. [1]

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 Greek Zeta archaic.svg  = 1, Greek Pi archaic.svg  = 5, Greek Delta 04.svg  = 10, Greek Eta classical.svg  = 100, Greek Chi normal.svg  = 1,000, and Greek Mu classical.svg  = 10,000. The numbers 50, 500, 5,000, and 50,000 were represented by the letter Greek Pi archaic.svg with minuscule powers of ten written in the top right corner: Attic 00050.svg , Attic 00500.svg , Attic 05000.svg , and Attic 50000.svg . [1] One-half was represented by 𐅁 (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 Greek Psi V-shaped.svg in Boeotia. [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-Eucleidean 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 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, [6] and its use is unattested in Athens until the 2nd century CE. [7] (In general, Athenians resisted using the new numerals for the longest of any Greek state, but had fully adopted them by c. 50 CE. [2] )

Description

Greek numerals in a c. 1100 Byzantine manuscript of Hero of Alexandria's Metrika. The first line contains the number ",thspqst d' st' ", i.e. "9,996 + .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px} 1/4 + 1/6". It features each of the special numeral symbols sampi (sp), koppa (q), and stigma (st) in their minuscule forms. Greek minuscule numerals Cod.Const.Pal.Vet.f96r.svg
Greek numerals in a c. 1100 Byzantine manuscript of Hero of Alexandria's Metrika. The first line contains the number "͵θϡϟϛ δʹ ϛʹ", i.e. "9,996 + 1⁄4 + 1⁄6". It features each of the special numeral symbols sampi (ϡ), koppa (ϟ), and stigma (ϛ) in their 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 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] (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 ])

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 Greek Sigma classical.svg Greek Mu classical.svg Greek Alpha classical.svg  (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. [9] ) In ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars: α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as 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, and γ°ʹ or γoʹ was two-thirds. These fractions were additive (also known as Egyptian fractions); for example δʹ ϛʹ indicated 1⁄4 + 1⁄6 = 5⁄12.

A 14th-century Byzantine map of the British Isles from a manuscript of Ptolemy's Geography, using Greek numerals for its graticule: 52-63degN of the equator and 6-33degE from Ptolemy's Prime Meridian at the Fortunate Isles. Add 19391 19-20.png
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 papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early.[ clarification needed ] 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 ( Greek Koppa cursive 02.svg and Greek Koppa cursive 03.svg ) and simplified ( Greek Koppa cursive 04.svg and Greek Koppa cursive 05.svg ). 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 ( Greek Digamma angular.svg ). By the Byzantine era, the letter was known as episemon and written as Greek Digamma cursive 02.svg or Greek Digamma cursive 06.svg . This eventually merged with the sigma-tau ligature stigma ϛ ( Greek Digamma cursive 07.svg or Greek Digamma cursive 04.svg ).

In modern Greek, a number of other changes have been made. Instead of extending an over bar over an entire number, the keraia (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 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 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 (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. [10]

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 isopsephy . Similar practices for the Hebrew and English are called gematria and English Qaballa, respectively.

Table

AncientByzantineModernValueAncientByzantineModernValueAncientByzantineModernValue
Greek Alpha classical.svg α̅Αʹ 1 Greek Iota classical.svg ι̅Ιʹ 10 Greek Rho classical.svg ρ̅Ρʹ 100
Greek Beta classical.svg β̅Βʹ 2 Greek Kappa classical.svg κ̅Κʹ 20 Greek Sigma classical.svg σ̅Σʹ 200
Greek Gamma classical.svg γ̅Γʹ 3 Greek Lambda classical.svg λ̅Λʹ 30 Greek Tau classical.svg τ̅Τʹ 300
Greek Delta classical.svg δ̅Δʹ 4 Greek Mu classical.svg μ̅Μʹ 40 Greek Upsilon classical.svg υ̅Υʹ 400
Greek Epsilon classical.svg ε̅Εʹ 5 Greek Nu classical.svg ν̅Νʹ 50 Greek Phi classical.svg φ̅Φʹ 500
Greek Digamma oblique.svg
Greek Digamma angular.svg 		Greek Digamma cursive 02.svg  and  Greek Digamma cursive 04.svg
Greek Digamma cursive 06.svg  and  Greek Digamma cursive 07.svg 		Ϛʹ


ΣΤʹ

 6 Greek Xi classical.svg ξ̅Ξʹ 60 Greek Chi classical.svg χ̅Χʹ 600
Greek Zeta classical.svg ζ̅Ζʹ 7 Greek Omicron classical.svg ο̅Οʹ 70 Greek Psi classical.svg ψ̅Ψʹ 700
Greek Eta classical.svg η̅Ηʹ 8 Greek Pi classical.svg π̅Πʹ 80 Greek Omega classical.svg ω̅Ωʹ 800
Greek Theta classical.svg θ̅Θʹ 9 Greek Koppa normal.svg
Greek Koppa cursive 01.svg 		Greek Koppa cursive 02.svg  and  Greek Koppa cursive 04.svg
Greek Koppa cursive 03.svg  and  Greek Koppa cursive 05.svg 		Ϟʹ 90 Greek Sampi Ionian.svg
Greek Sampi palaeographic 05.svg  and  Greek Sampi palaeographic 15.svg
Greek Sampi palaeographic 06.svg  and  Greek Sampi palaeographic 09.svg 		Greek Sampi palaeographic 03.svg  and  Greek Sampi palaeographic 07.svg
Greek Sampi palaeographic 08.svg
Greek Sampi palaeographic 10.svg  and  Greek Sampi palaeographic 11.svg
Greek Sampi palaeographic 14.svg  and  Greek Sampi palaeographic 13.svg
Sampi.svg 		Ϡʹ 900
Greek Sampi 1000.svg  and  Greek Sampi 1000 (2).svg ͵α͵Α 1000 Greek Iota classical.svg Greek Sampi palaeographic 02.svg ͵ι͵Ι 10000 Greek Rho classical.svg Greek Sampi palaeographic 02.svg ͵ρ͵Ρ 100000
Greek Beta classical.svg Greek Sampi palaeographic 02.svg ͵β͵Β 2000 Greek Kappa classical.svg Greek Sampi palaeographic 02.svg ͵κ͵Κ 20000 Greek Sigma classical.svg Greek Sampi palaeographic 02.svg ͵σ͵Σ 200000
Greek Gamma classical.svg Greek Sampi palaeographic 02.svg ͵ Greek Gamma 02.svg ,Γ 3000 Greek Lambda classical.svg Greek Sampi palaeographic 02.svg ͵λ͵Λ 30000 Greek Tau classical.svg Greek Sampi palaeographic 02.svg ͵τ͵Τ 300000
Greek Delta classical.svg Greek Sampi palaeographic 02.svg ͵ Greek Delta classical.svg ͵Δ 4000 Greek Mu classical.svg Greek Sampi palaeographic 02.svg ͵μ͵Μ 40000 Greek Upsilon classical.svg Greek Sampi palaeographic 02.svg ͵υ͵Υ 400000
Greek Epsilon classical.svg Greek Sampi palaeographic 02.svg ͵ε͵Ε 5000 Greek Nu classical.svg Greek Sampi palaeographic 02.svg ͵ν͵Ν 50000 Greek Phi classical.svg Greek Sampi palaeographic 02.svg ͵φ͵Φ 500000
Greek Digamma oblique.svg Greek Sampi palaeographic 02.svg
Greek Digamma angular.svg Greek Sampi palaeographic 02.svg 		͵ Greek Digamma cursive 02.svg  and ͵ Greek Digamma cursive 04.svg
͵ Greek Digamma cursive 06.svg  and ͵ Greek Digamma cursive 07.svg 		͵Ϛ
͵ΣΤ		 6000 Greek Xi classical.svg Greek Sampi palaeographic 02.svg ͵ξ͵Ξ 60000 Greek Chi classical.svg Greek Sampi palaeographic 02.svg ͵χ͵Χ 600000
Greek Zeta classical.svg Greek Sampi palaeographic 02.svg ͵ζ͵Ζ 7000 Greek Omicron classical.svg Greek Sampi palaeographic 02.svg ͵ο͵Ο 70000 Greek Psi classical.svg Greek Sampi palaeographic 02.svg ͵ψ͵Ψ 700000
Greek Eta classical.svg Greek Sampi palaeographic 02.svg ͵η͵Η 8000 Greek Pi classical.svg Greek Sampi palaeographic 02.svg ͵π͵Π 80000 Greek Omega classical.svg Greek Sampi palaeographic 02.svg ͵ω͵Ω 800000
Greek Sampi 9000.svg ͵θ͵Θ 9000 Greek Koppa normal.svg Greek Sampi palaeographic 02.svg
Greek Koppa cursive 01.svg Greek Sampi palaeographic 02.svg 		͵ Greek Koppa cursive 02.svg  and ͵ Greek Koppa cursive 04.svg
͵ Greek Koppa cursive 03.svg  and ͵ Greek Koppa cursive 05.svg 		͵Ϟ 90000 Greek Sampi Ionian.svg Greek Sampi palaeographic 02.svg
Greek Sampi palaeographic 05.svg Greek Sampi palaeographic 02.svg  and  Greek Sampi palaeographic 15.svg Greek Sampi palaeographic 02.svg
Greek Sampi palaeographic 06.svg Greek Sampi palaeographic 02.svg  and  Greek Sampi palaeographic 09.svg Greek Sampi palaeographic 02.svg 		͵ Greek Sampi palaeographic 03.svg  and ͵ Greek Sampi palaeographic 07.svg
͵ Greek Sampi palaeographic 08.svg
͵ Greek Sampi palaeographic 10.svg  and ͵ Greek Sampi palaeographic 11.svg
͵ Greek Sampi palaeographic 14.svg  and ͵ Greek Sampi palaeographic 13.svg
͵ Sampi.svg 		͵Ϡ 900000

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; αΜ was 10,000, βΜ was 10,0002 = 100,000,000, γΜ was 10,0003 = 1012 and so on. [11]

Zero

Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus P. Lund, Inv. 35a.jpg
Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus

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 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 c. 140 BC. It was then used by Ptolemy (c. 140 BC), Theon (c. 380  AD) and Theon's daughter Hypatia (d. 415 AD). The symbol for zero is clearly different from that of the value for 70, omicron or "ο". 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:

Each number in the first column, labeled περιφερειῶν, ["regions"] is the number of degrees of arc on a circle. Each number in the second column, labeled εὐθειῶν, ["straight lines" or "segments"] 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 84+1⁄2°. In the next column we see π μα γ , meaning   80 + ⁠41/60⁠ + ⁠3/60²⁠. That is the length of the chord corresponding to an arc of 84+1⁄2° when the diameter of the circle is 120. The next column, labeled 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 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, 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 manuscripts, leaving a bare ο (omicron). [12] [13] This gradual change from an invented symbol to ο does not support the hypothesis that the latter was the initial of οὐδέν meaning "nothing". [14] [15] 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 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. [16]

The Greek zero was added to Unicode[ when? ] at U+1018A𐆊GREEK ZERO SIGN.

See also

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References

  1. 1 2 Verdan, Samuel (20 March 2007). "Systèmes numéraux en Grèce ancienne: Description et mise en perspective historique" (in French). Archived from the original on 2 February 2010. Retrieved 2 March 2011.
  2. 1 2 Heath, Thomas L. (2003) [1931]. A Manual of Greek Mathematics ([2003] reprint ed.). Oxford, UK: Oxford University Press[1931]; Dover Books[2003]. pp.  14 ff. ISBN   9780486154442 . Retrieved 1 November 2013– via Google Books.
  3. ↑ Thompson, Edward M. (1893). Handbook of Greek and Latin Palaeography. New York, NY: D. Appleton. p. 114.
  4. ↑ "IG I³ 1387". Searchable Greek Inscriptions. The Packard Humanities Institute. Cornell University & Ohio State University. IG I³ 1387 also known as IG I² 760. Retrieved 1 November 2013.
  5. ↑ Jeffery, Lilian H. (1961). The Local Scripts of Archaic Greece. Oxford, UK: Clarendon Press. pp. 38 ff.
  6. ↑ "Magnesia 4". Searchable Greek Inscriptions. The Packard Humanities Institute. Cornell University & Ohio State University. Magnesia 4 also known as Syll³ 695.b. Retrieved 1 November 2013.
  7. ↑ "IG II² 2776". Searchable Greek Inscriptions. The Packard Humanities Institute. Cornell University & Ohio State University . Retrieved 1 November 2013.
  8. ↑ Edkins, Jo (2006). "Classical Greek Numbers". Archived from the original on 10 May 2013. Retrieved 29 April 2013.
  9. ↑ Heath, Thomas L. A Manual of Greek Mathematics, pp. 14 ff. Oxford Univ. Press (Oxford), 1931. Reprinted Dover (Mineola), 2003. Accessed 1 November 2013.
  10. ↑ Nick Nicholas (9 April 2005). "Numerals: Stigma, Koppa, Sampi". Archived from the original on 5 August 2012. Retrieved 2 March 2011.
  11. 1 2 Greek number systems - MacTutor
  12. ↑ https://www.unicode.org/L2/L2004/04054-greek-zero.pdf [ bare URL PDF ]
  13. ↑ https://raymondm.co.uk/prog/GreekZeroSign.pdf [ bare URL PDF ]
  14. ↑ Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity (2, reprint ed.). Dover Publications. pp. 13–14, plate 2. ISBN   978-0-486-22332-2.
  15. ↑ Mercier, Raymond. "Consideration of the Greek symbol 'zero'" (PDF). — gives numerous examples
  16. ↑ Ptolemy, Claudius (1998) [100–170 AD]. "Book VI". Ptolemy's Almagest . Translated by Toomer, G.J. Princeton, NJ: Princeton University Press. pp. 306–307.

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