Greek numerals

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.

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:  = 1,  = 10,  = 100,  = 1,000, and  = 10,000.

thumb|[[Tribute list using attic numerals, 430s BC.]]

In the classical times two different systems of notation were used side by side: Firstly, an acrophonic system called Attic numerals -- also known as Herodianic signs -- and, secondly, alphabetic numerals.

The Attic numerals ran from = 1, = 5, = 10, = 100, = 1,000, and = 10,000.

These were repeated at most four times; so, 5, 50, 500, and 5,000 represented by the letter with minuscule powers of ten written in the top-right corner: x16px|class=skin-invert, x16px|class=skin-invert, x16px|class=skin-invert, and x16px|class=skin-invert

This system was in limited use from at least 350 BC, but the first official use of the alphabetic system in Greece proper is from c.a. 200 BC.

More thorough modern archaeology has caused the date to be pushed back at least to the 5th century BC, 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. 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, and its use is unattested in Athens until the 2nd century CE. (In general, Athenians resisted using the new numerals for the longest of any Greek state, but had fully adopted them by . Because the Phoenician precursor of sampi fell between what came to represent pi and koppa in Greek, it's use to represent 900 is taken as a proof that the sampi had been fallen into disuse as a letter by the time the system was created.

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  (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.) In ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars: , , , etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as  (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 (); γ indicated one third, δ one fourth and so on. As an exception, special symbol ∠ indicated one half, and γ° or γo was two-thirds. These fractions were additive (also known as Egyptian fractions); for example indicated .

thumb|left|300px|A 14th-century Byzantine map of the British Isles from a manuscript of [[Claudius Ptolemy|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. 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 (x16px|class=skin-invert and x16px|class=skin-invert) and simplified (x16px|class=skin-invert and x16px|class=skin-invert). 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 the Byzantine era, the letter was known as episemon and written as or . This eventually merged with the sigma-tau ligature stigma ϛ ( or ).

In modern Greek, a number of other changes have been made. Instead of extending an over bar over an entire number, the keraia (, 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 in modern Greek. A lower left keraia (Unicode: U+0375, "Greek Lower Numeral Sign") is now standard for distinguishing thousands: 2019 is represented as ΒΙΘ ().

The declining use of ligatures in the 20th century also means that stigma is frequently written as the separate letters ΣΤ, although a single keraia is used for the group.

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

{| class="wikitable" style="text-align:center;margin-left:auto;margin-right:auto;"

|+ Historical number forms using Greek letters

! Ancient !! Byzantine !! Modern !! Value

| || α̅ || || 1

| || β̅ || || 2

| || γ̅ || || 3

| || δ̅ || || 4

| || ε̅ || || 5

| ||  and   and  || ΣΤ || 6

| || ζ̅ || || 7

| || η̅ || || 8

| || θ̅ || || 9

| || ι̅ || || 10

| || κ̅ || || 20

| || λ̅ || || 30

| || μ̅ || || 40

| || ν̅ || || 50

| || ξ̅ || Ξ || 60

| || ο̅ || || 70

| || π̅ || || 80

| ||  and   and  || Ϟ || 90

| || ρ̅ || || 100

| || σ̅ || || 200

| || τ̅ || Τ || 300

| || υ̅ || || 400

| || φ̅ || || 500

| || χ̅ || || 600

| || ψ̅ || || 700

| || ω̅ || || 800

|  and   and  ||  and   and   and  class=skin-invert-image|x16px || Ϡ || 900

|  and  || α || Α || 1000

| class=skin-invert-image|x8pxclass=skin-invert-image|x16px || β || Β || 2000

| class=skin-invert-image|x8pxclass=skin-invert-image|x16px || || Γ || 3000

| class=skin-invert-image|x8pxclass=skin-invert-image|x16px || || Δ || 4000

| class=skin-invert-image|x8pxclass=skin-invert-image|x16px || ε || Ε || 5000

| class=skin-invert-image|x8pxclass=skin-invert-image|x16px class=skin-invert-image|x8pxclass=skin-invert-image|x16px ||  and   and  || Ϛ ΣΤ || 6000

| class=skin-invert-image|x8pxclass=skin-invert-image|x16px || ζ||Ζ || 7000

| class=skin-invert-image|x8pxclass=skin-invert-image|x16px || η || Η || 8000

| || θ || Θ || 9000

Sub-sections of Greek manuscripts are sometimes numbered by lowercase characters (α. β. γ. δ. ε. ϛ. ζ. η. θ.).
In Ancient Greek, myriad notation is used for multiples of 10,000, for example for 20,000 or for 1,234,567 (also written on the line as Μ ).

Higher numbers

In his text The 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 an 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 a new enumeration scheme with much greater range than any of those shown above.

Pappus of Alexandria reports that Apollonius of Perga developed a simpler system based on powers of the myriad:

: &emsp; was   10,000

: &emsp; was   (10,000)2 100,000,000

: &emsp; was   (10,000)3

and so on. This gradual change from an invented symbol to 'ο' does not support the hypothesis that the omicron was the initial of 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 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 , 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.

The Greek zero was added to Unicode in Version 4.1.0 at .

References

External links

The Greek Number Converter