 | This article'slead sectionmay be too short to adequatelysummarize the key points. Please consider expanding the lead toprovide an accessible overview of all important aspects of the article.(March 2024) |
| Part ofa series on | ||||
| Numeral systems | ||||
|---|---|---|---|---|
| ||||
| List of numeral systems | ||||
Greek numerals, also known asIonic,Ionian,Milesian, orAlexandrian numerals, is asystem of writing numbers using the letters of theGreek alphabet. In modernGreece, they are still used forordinal numbers and in contexts similar to those in whichRoman numerals are still used in theWestern world. For ordinarycardinal numbers, however, modern Greece usesArabic numerals.
TheMinoan andMycenaean civilizations'Linear A andLinear B alphabets used a different system, calledAegean numerals, which included number-only symbols for powers of ten:𐄇 = 1,𐄐 = 10,𐄙 = 100,𐄢 = 1,000, and𐄫 = 10,000.[1]
Attic numerals composed another system that came into use perhaps in the 7th century BC. They wereacrophonic, 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 by𐅁 (left half of a full circle) and one-quarter by ɔ (right side of a full circle). The same system was used outside ofAttica, but the symbols varied with thelocal alphabets; for example, 1,000 was
inBoeotia.[2]
The present system probably developed aroundMiletus inIonia. 19th century classicists placed its development in the 3rd century BC, the occasion of its first widespread use.[3] More thoroughmodern archaeology has caused the date to be pushed back at least to the 5th century BC,[4] a little beforeAthens abandoned itspre-Eucleidean alphabet in favour ofMiletus's in 402 BC, and it may predate that by a century or two.[5] The present system uses the 24 letters adopted underEucleides, as well as threePhoenician and Ionic ones that had not been dropped from the Athenian alphabet (although kept for numbers):digamma,koppa, andsampi. 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 forsampi, 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 byc. 50 CE.[2])
Greek numerals aredecimal, based on powers of 10. The units from 1 to 9 are assigned to the first nine letters of the oldIonic alphabet fromalpha totheta. 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 fromiota tokoppa. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well, fromrho tosampi.[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])
Thisalphabetic 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
(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 inAsia Minor well into theRoman period.[9]) In ancient and medieval manuscripts, these numerals were eventually distinguished from letters usingoverbars:α,β,γ, etc. In medieval manuscripts of theBook of Revelation, thenumber of the Beast 666 is written asχξϛ (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 akeraia (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 asEgyptian fractions); for exampleδʹ ϛʹ indicated1⁄4 +1⁄6 =5⁄12.
Although theGreek alphabet began with onlymajuscule forms, survivingpapyrus manuscripts fromEgypt show thatuncial andcursiveminuscule 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 (
and
) and simplified (
and
). 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 (
). By theByzantine era, the letter was known asepisemon and written as
or
. This eventually merged with thesigma-tauligaturestigma ϛ (
or
).
Inmodern Greek, a number of other changes have been made. Instead of extending an over bar over an entire number, thekeraia (κεραία,lit. "hornlike projection") is marked to its upper right, a development of the short marks formerly used for single numbers and fractions. The modernkeraia (ʹ) is a symbol similar to theacute accent (´), thetonos (U+0384,΄) and the prime symbol (U+02B9, ʹ), but has its ownUnicode character as U+0374.Alexander the Great's fatherPhilip II of Macedon is thus known asΦίλιππος Βʹ in modern Greek. A lower leftkeraia (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 singlekeraia is used for the group.[10]
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 calledisopsephy. Similar practices for the Hebrew and English are calledgematria andEnglish Qaballa, respectively.
| Ancient | Byzantine | Modern | Value |
|---|---|---|---|
| α̅ | Αʹ | 1 |
 | β̅ | Βʹ | 2 |
 | γ̅ | Γʹ | 3 |
 | δ̅ | Δʹ | 4 |
 | ε̅ | Εʹ | 5 |
 | and and | Ϛʹ ΣΤʹ | 6 |
 | ζ̅ | Ζʹ | 7 |
| η̅ | Ηʹ | 8 |
 | θ̅ | Θʹ | 9 |
 | ι̅ | Ιʹ | 10 |
 | κ̅ | Κʹ | 20 |
 | λ̅ | Λʹ | 30 |
| μ̅ | Μʹ | 40 |
 | ν̅ | Νʹ | 50 |
 | ξ̅ | Ξʹ | 60 |
 | ο̅ | Οʹ | 70 |
 | π̅ | Πʹ | 80 |
  | and and | Ϟʹ | 90 |
 | ρ̅ | Ρʹ | 100 |
| σ̅ | Σʹ | 200 |
 | τ̅ | Τʹ | 300 |
 | υ̅ | Υʹ | 400 |
 | φ̅ | Φʹ | 500 |
 | χ̅ | Χʹ | 600 |
 | ψ̅ | Ψʹ | 700 |
 | ω̅ | Ωʹ | 800 |
  and   and  |  and    and   and   | Ϡʹ | 900 |
 and .svg) | ͵α | ͵Α | 1000 |
 | ͵β | ͵Β | 2000 |
| ͵ | ͵Γ | 3000 |
| ͵ | ͵Δ | 4000 |
| ͵ε | ͵Ε | 5000 |
| ͵ and ͵ ͵ and ͵ | ͵Ϛ ͵ΣΤ | 6000 |
| ͵ζ | ͵Ζ | 7000 |
| ͵η | ͵Η | 8000 |
 | ͵θ | ͵Θ | 9000 |
In his textThe Sand Reckoner,Archimedes gives an upper bound of the number of grains of sand required to fill the entire universe, using an estimate of its size current in his time. His essay demonstrated a easily visualized contradiction of the then-held adage that it was impossible to name a quantity "greater than the number of grains of sand on a beach", or in the entire world. In order to do that, he devised anew enumeration scheme with much greater range than any of those shown above.
Pappus of Alexandria reports thatApollonius of Perga developed a simpler system based onpowers of the myriad:
and so on.[11]
Hellenisticastronomers extended alphabetic Greek numerals into asexagesimalpositionalnumbering system by limiting each position to a maximum value of 50 + 9 which uses only the letters up throughnu (ν) and included otherwise unusedomicron (ο) as a special symbol forzero. Omicron as zero 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 fromBabylonian numerals byHipparchusc. 140 BCE. It was then used byPtolemy (c. 140 BCE),Theon (c. 380 CE) and Theon's daughterHypatia (d. 415 CE). The symbolomicron or'ο' as used for zero in astronomical and mathematical data tables is clearly different from its conventional use as the value for 70. 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.
InPtolemy's table of chords, the first fairly extensivetrigonometric table, there were 360 rows, portions of which looked as follows:
Each number in the first column, labeledπεριφερειῶν, ("regions") is the number of degrees of arc on a circle. Each number in the second column, labeledεὐθειῶν, ("straight lines" or "segments") is the length of the corresponding chord of the circle, when the diameter is 120 units. Thusπδ represents an 84° arc, and the∠′ after it means one-half, so thatπδ∠′ means84+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 of84+1/2° when the diameter of the circle is 120 units. The next column, labeledἑξηκοστῶν, 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 forlinear interpolation.
The Greeksexagesimal placeholder or zero symbol changed over time: The symbol used onpapyri during the second century was a very small circle with an overbar several diameters long ( ο ), terminated or not at both ends in various ways. Later, the overbar shortened to only one letter-diameter, similar to the modern 'o'+macron (ō). It was still being used in late medieval Arabic manuscripts whenever alphabetic numerals were used. In laterByzantine manuscripts even the minimal overbar was omitted, leaving a bare 'ο' (omicron).[12][13] This gradual change from an invented symbol ο to 'ο' does not support the hypothesis that the omicron was the initial ofοὐδέν meaning "nothing". 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 asexagesimal 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 theMoon and either the center of theSun (forsolar eclipses) or the center ofEarth's shadow (forlunar eclipses). All of these zeros took the formο | 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 asdigits (of five arc-minutes each), whereas the fractional part was in the next column labeledminute of immersion, meaning sixtieths (and thirty-six-hundredths) of a digit.[14]
The Greek zero was added to Unicode in Version 4.1.0 atU+1018A 𐆊GREEK ZERO SIGN.[15]
| |||||||||||||||||||||||||
| |||||||||||||||||||||||||
| |||||||||||||||||||||||||
| |||||||||||||||||||||||||
| |||||||||||||||||||||||||
| |||||||||||||||||||||||||
