Greek numerals

System of writing numbers

The numerical signs ʹ and ͵ redirect here. For the accent mark ', see Acute accent.

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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

in Boeotia

.[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 1⁄4 + 1⁄6 = 5⁄12.

Prime Meridian at the Fortunate Isles

.

stigma ϛ (

or

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. 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	,Α	1000			͵ι	,Ι	10000			͵ρ	,Ρ	100000
	͵β	,Β	2000			͵κ	,Κ	20000			͵σ	,Σ	200000
	͵	,Γ	3000			͵λ	,Λ	30000			͵τ	,Τ	300000
	͵	,Δ	4000			͵μ	,Μ	40000			͵υ	,Υ	400000
	͵ε	,Ε	5000			͵ν	,Ν	50000			͵φ	,Φ	500000
	͵	,Ϛ ,Ϝ ,ΣΤ	6000			͵ξ	,Ξ	60000			͵χ	,Χ	600000
	͵ζ	,Ζ	7000			͵ο	,Ο	70000			͵ψ	,Ψ	700000
	͵η	,Η	8000			͵π	,Π	80000			͵ω	,Ω	800000
	͵θ	,Θ	9000			͵	,Ϟ ,Ϙ	90000			͵	,Ϡ ,Ͳ	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 βΜ for 20,000 or ρκγΜ͵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; 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

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+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 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]

See also

• Alphabetic numeral system – Type of numeral system
• Attic numerals – Symbolic number notation used by the ancient Greeks
• Cyrillic numerals – Numeral system derived from the Cyrillic script
• Greek mathematics – Mathematics of Ancient Greeks
• Greek numerals in Unicode
   – Graphemes for various number systemsPages displaying short descriptions of redirect targets (acrophonic, not alphabetic, numerals)
• Hebrew numerals – Numeral system using letters of the Hebrew alphabet, based on the Greek system
• History of ancient numeral systems – Symbols representing numbers
• History of arithmetic – Aspect of history
• History of communication – Aspect of history
• Isopsephy – Practice of adding up number values of letters in a word to form a single number
• List of numeral system topics
• List of numeral systems
• Number of the beast – Number associated with the Beast of Revelation
• Roman numerals – Numbers in the Roman numeral system

References

External links

Wikimedia Commons has media related to Greek numerals.

• The Greek Number Converter