Greek numerals

Description

Greek numerals form a sign-value notation system in decimal. The units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. 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 from iota to koppa. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well, from rho to sampi. Because the Phoenician precursor of sampi fell between what came to represent pi and koppa in Greek, its use to represent 900 is taken as a proof that the sampi had 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 BCE 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: 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.

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 ( 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 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 (κεραία, 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 (ʹ) 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 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 (2,000 + 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 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

• Sub-sections of Greek 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 for 1,234,567 (also written on the line as ρκγΜ ͵δφξζ).

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:

was 10,000 = 10⁴

was (10,000)² = 100,000,000 = 10⁸

was (10,000)³ = 10¹²

and so on.

Zero

As a non-positional system, Greek numerals do not have an official representation of the number zero.

Hellenistic astronomers extended alphabetic Greek numerals into a sexagesimal positional numbering system by limiting each position to a maximum value of 50 + 9 which uses only the letters up through nu (ν) and included otherwise unused omicron (ο) as a special symbol for zero. 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 from Babylonian numerals by Hipparchus c. 140 BCE. It was then used by Ptolemy (c. 140 BCE), Theon (c. 380 CE) and Theon's daughter Hypatia (d. 415 CE). The symbol omicron 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.

In Ptolemy's table of chords, the first fairly extensive trigonometric table, there were 360 rows, portions of which looked as follows:

${\displaystyle \begin{array}{ccc}\pi \epsilon \varrho \iota \phi \epsilon \varrho \epsilon \iota \tilde{\omega }\nu & \epsilon \stackrel{\text{'}}{\upsilon }\vartheta \epsilon \iota \tilde{\omega }\nu & \stackrel{\text{‘}}{\epsilon }\xi \eta \kappa o\sigma \tau \tilde{\omega }\nu \\ \begin{array}{|l|}\hline \pi \delta {\mathrm{\angle }}^{\prime }\\ \pi \epsilon \\ \pi \epsilon {\mathrm{\angle }}^{\prime }\\ \pi ϛ\\ \pi ϛ{\mathrm{\angle }}^{\prime }\\ \pi \zeta \\ \hline\end{array}& \begin{array}{|rrr|}\hline \pi & \mu \alpha & \gamma \\ \pi \alpha & \delta & \iota \epsilon \\ \pi \alpha & \kappa \zeta & \kappa \beta \\ \pi \alpha & \nu & \kappa \delta \\ \pi \beta & \iota \gamma & \iota \vartheta \\ \pi \beta & \lambda ϛ& \vartheta \\ \hline\end{array}& \begin{array}{|rrrr|}\hline \circ & \circ & \mu ϛ& \kappa \epsilon \\ \circ & \circ & \mu ϛ& \iota \delta \\ \circ & \circ & \mu ϛ& \gamma \\ \circ & \circ & \mu \epsilon & \nu \beta \\ \circ & \circ & \mu \epsilon & \mu \\ \circ & \circ & \mu \epsilon & \kappa \vartheta \\ \hline\end{array}\end{array}}$

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 units. 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 units. The next column, labeled ἑξηκοστῶν, 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 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 ( 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 later Byzantine manuscripts even the minimal overbar was omitted, leaving a bare 'ο' (omicron). This gradual change from an invented symbol ο to 'ο' does not support the hypothesis that the omicron was the initial of οὐδέν 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 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.

The Greek zero was added to Unicode in Version 4.1.0 at U+1018A 𐆊 GREEK ZERO SIGN.