Greek numerals

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

acrophonic, derived (after the initial one) from the first letters of the names of the numbers represented. They ran  = 1,  = 5,  = 10,  = 100,  = 1,000, and  = 10,000. The numbers 50, 500, 5,000, and 50,000 were represented by the letter with minuscule powers of ten written in the top right corner: , , , and .^[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 in Boeotia.^[2]

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

Table

Ancient	Byzantine	Modern	Value		Ancient	Byzantine	Modern	Value		Ancient	Byzantine	Modern	Value
	α̅	Αʹ	1			ι̅	Ιʹ	10			ρ̅	Ρʹ	100
	β̅	Βʹ	2			κ̅	Κʹ	20			σ̅	Σʹ	200
	γ̅	Γʹ	3			λ̅	Λʹ	30			τ̅	Τʹ	300
	δ̅	Δʹ	4			μ̅	Μʹ	40			υ̅	Υʹ	400
	ε̅	Εʹ	5			ν̅	Νʹ	50			φ̅	Φʹ	500
	and   and	Ϛʹ Ϝʹ ΣΤʹ	6			ξ̅	Ξʹ	60			χ̅	Χʹ	600
	ζ̅	Ζʹ	7			ο̅	Οʹ	70			ψ̅	Ψʹ	700
	η̅	Ηʹ	8			π̅	Πʹ	80			ω̅	Ωʹ	800
	θ̅	Θʹ	9			and   and	Ϟʹ Ϙʹ	90		and   and	and   and   and	Ϡʹ Ͳʹ	900
and	͵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
	͵ and ͵ ͵ and ͵	,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			͵ and ͵ ͵ and ͵	,Ϟ ,Ϙ	90000		and   and	͵ and ͵ ͵ ͵ and ͵ ͵ and ͵ ͵	,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

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, ["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 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
   – Branch of elementary mathematicsPages displaying short descriptions of redirect targets
• History of communication
• Isopsephy – Greek numerological practice
• 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