# Greek numerals

Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a

cardinal numbers, however, modern Greece uses Arabic numerals
.

## History

The

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]

acrophonic, derived (after the initial one) from the first letters of the names of the numbers represented. They ran .[2]

The present system probably developed around

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 BCE,[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

stigma (ϛ) in their minuscule
forms.

Greek numerals are

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

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 14 + 16 = 512.

Although the

stigma ϛ (
or
).

In

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).

## 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. A similar practice adapted for the Hebrew alphabet is referred to as gematria.

## Table

Ancient Byzantine Modern Value Ancient Byzantine Modern Value Ancient Byzantine Modern Value
α Αʹ
1
ι Ιʹ
10
ρ Ρʹ
100
β Βʹ
2
κ Κʹ 20
σ Σʹ 200
γ Γʹ
3
λ Λʹ 30
τ Τʹ 300
δ Δʹ
4
μ Μʹ 40
υ Υʹ 400
ε Εʹ
5
ν Νʹ 50
φ Φʹ 500
Ϛʹ
Ϝʹ
ΣΤʹ
6
ξ Ξʹ 60
χ Χʹ 600
ζ Ζʹ
7
ο Οʹ 70
ψ Ψʹ 700
η Ηʹ
8
π Πʹ 80
ω Ωʹ 800
θ Θʹ
9
Ϟʹ
Ϙʹ
90
Ϡʹ
Ͳʹ
900
͵α 1000
͵ι
10000
͵ρ
100000
͵β 2000
͵κ
20000
͵σ
200000
͵
3000
͵λ
30000
͵τ
300000
͵
4000
͵μ
40000
͵υ
400000
͵ε 5000
͵ν
50000
͵φ
500000
͵

,ΣΤ
6000
͵ξ
60000
͵χ
600000
͵ζ 7000
͵ο
70000
͵ψ
700000
͵η 8000
͵π
80000
͵ω
800000
͵θ 9000
͵

90000
͵

900000
• 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 for 20,000 or ͵δφξζ (also written on the line as ρκγΜ ͵δφξζ) for 1,234,567.[11]

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

## Zero

Hypatia (died 415). 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:

${\displaystyle {\begin{array}{ccc}\pi \varepsilon \varrho \iota \varphi \varepsilon \varrho \varepsilon \iota {\tilde {\omega }}\nu &\varepsilon {\overset {\text{'}}{\upsilon }}\vartheta \varepsilon \iota {\tilde {\omega }}\nu &{\overset {\text{`}}{\varepsilon }}\xi \eta \kappa \mathrm {o} \sigma \tau {\tilde {\omega }}\nu \\{\begin{array}{|l|}\hline \pi \delta \angle '\\\pi \varepsilon \\\pi \varepsilon \angle '\\\hline \pi \mathrm {\stigma} \\\pi \mathrm {\stigma} \angle '\\\pi \zeta \\\hline \end{array}}&{\begin{array}{|r|r|r|}\hline \pi &\mu \alpha &\gamma \\\pi \alpha &\delta &\iota \varepsilon \\\pi \alpha &\kappa \zeta &\kappa \beta \\\hline \pi \alpha &\nu &\kappa \delta \\\pi \beta &\iota \gamma &\iota \vartheta \\\pi \beta &\lambda \mathrm {\stigma} &\vartheta \\\hline \end{array}}&{\begin{array}{|r|r|r|r|}\hline \circ &\circ &\mu \mathrm {\stigma} &\kappa \varepsilon \\\circ &\circ &\mu \mathrm {\stigma} &\iota \delta \\\circ &\circ &\mu \mathrm {\stigma} &\gamma \\\hline \circ &\circ &\mu \varepsilon &\nu \beta \\\circ &\circ &\mu \varepsilon &\mu \\\circ &\circ &\mu \varepsilon &\kappa \vartheta \\\hline \end{array}}\end{array}}}$

Each number in the first column, labeled περιφερειῶν, is the number of degrees of arc on a circle. Each number in the second column, labeled 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 84+12°. 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+12° 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 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".[12][13] 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.[14]

Character information
Preview 𐆊
Unicode name GREEK ZERO SIGN
Encodings decimal hex
Unicode 65930 U+1018A
UTF-8 240 144 134 138 F0 90 86 8A
UTF-16 55296 56714 D800 DD8A
Numeric character reference &#65930; &#x1018A;

## References

1. ^ a b 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. ^ Thompson, Edward M. (1893). Handbook of Greek and Latin Palaeography. New York, NY: D. Appleton. p. 114.
3. ^ "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.
4. ^ Jeffery, Lilian H. (1961). The Local Scripts of Archaic Greece. Oxford, UK: Clarendon Press. pp. 38 ff.
5. ^ "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.
6. ^ "IG II² 2776". Searchable Greek Inscriptions. The Packard Humanities Institute. Cornell University & Ohio State University. Retrieved 1 November 2013.
7. ^ Edkins, Jo (2006). "Classical Greek Numbers". Archived from the original on 10 May 2013. Retrieved 29 April 2013.
8. ^ Heath, Thomas L. A Manual of Greek Mathematics, pp. 14 ff. Oxford Univ. Press (Oxford), 1931. Reprinted Dover (Mineola), 2003. Accessed 1 November 2013.
9. ^ Nick Nicholas (9 April 2005). "Numerals: Stigma, Koppa, Sampi". Archived from the original on 5 August 2012. Retrieved 2 March 2011.
10. ^ a b Greek number systems - MacTutor
11. ^ .
12. ^ Mercier, Raymond. "Consideration of the Greek symbol 'zero'" (PDF). — gives numerous examples
13. ^
Toomer, G.J.
Princeton, NJ: Princeton University Press. pp. 306–307.