Alphabetic numeral system
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An alphabetic numeral system is a type of
History
The first attested alphabetic numeral system is the Greek alphabetic system (named the Ionic or Milesian system due to its origin in west Asia Minor). The system's structure follows the structure of the Egyptian demotic numerals; Greek letters replaced Egyptian signs. The first examples of the Greek system date back to the 6th century BC, written with the letters of the archaic Greek script used in Ionia.[2]
Other cultures in contact with Greece adopted this numerical notation, replacing the Greek letters with their own script; these included the Hebrews in the late 2nd century BC. The
The Arabs developed their own alphabetic numeral system, the abjad numerals, in the 7th century AD, and used it for mathematical and astrological purposes even as late as the 13th century far after the introduction of the Hindu–Arabic numeral system.[6] After the adoption of Christianity, Armenians and Georgians developed their alphabetical numeral system in the 4th or early 5th century, while in the Byzantine Empire Cyrillic numerals and Glagolitic were introduced in the 9th century. Alphabetic numeral systems were known and used as far north as England, Germany, and Russia, as far south as Ethiopia, as far east as Persia, and in North Africa from Morocco to Central Asia.[citation needed]
By the 16th century AD, most alphabetic numeral systems had died out or were in little use, displaced by Arabic positional and Western numerals as the ordinary numerals of commerce and administration throughout Europe and the Middle East.[1]
The newest alphabetic numeral systems in use, all of them positional, are part of tactile writing systems for
Systems
An alphabetic numeral system employs the letters of a script in the specific order of the alphabet in order to express numerals.
In Greek, letters are assigned to respective numbers in the following sets: 1 through 9, 10 through 90, 100 through 900, and so on. Decimal places are represented by a single symbol. As the alphabet ends, higher numbers are represented with various multiplicative methods. However, since writing systems have a differing number of letters, other systems of writing do not necessarily group numbers in this way. The
These various systems do not have a single unifying trait or feature. The most common structure is ciphered-additive with a decimal base, with or without the use of multiplicative-additive structuring for the higher numbers. Exceptions include the Armenian notation of Shirakatsi, which is multiplicative-additive and sometimes uses a base 1,000, and the Greek and Arabic astronomical notation systems.
Numeral signs
The tables below show the alphabetic numeral configurations of various writing systems.
Greek alphabetic numerals – "Ionian" or "Milesian numerals" – (minuscule letters)
units α β γ δ ε ϛ ζ η θ 1 2 3 4 5 6 7 8 9 tens ι κ λ μ ν ξ ο π ϟ 10 20 30 40 50 60 70 80 90 hundreds ρ σ τ υ φ χ ψ ω ϡ 100 200 300 400 500 600 700 800 900 thousands ͵α ͵β ͵γ ͵δ ͵ε ͵ϛ ͵ζ ͵η ͵θ 1000 2000 3000 4000 5000 6000 7000 8000 9000
Some numbers represented with Greek alphabetic numerals:
- ͵γϡμβ = (3000 + 900 + 40 + 2) = 3942
- χξϛ = (600 + 60 + 6) = 666
units א ב ג ד ה ו ז ח ט 1 2 3 4 5 6 7 8 9 tens י כ ל מ נ ס ע פ צ 10 20 30 40 50 60 70 80 90 hundreds ק ר ש ת 100 200 300 400 thousands 'א 'ב 'ג 'ד 'ה 'ו 'ז 'ח 'ט 1000 2000 3000 4000 5000 6000 7000 8000 9000
The Hebrew writing system has only twenty-two consonant signs, so numbers can be expressed with single individual signs only up to 400. Higher hundreds – 500, 600, 700, 800, and 900 – can be written only with various cumulative-additive combinations of the lower hundreds (direction of writing is right to left):[7]
- תק = (400+100) 500
- תר = (400+200) 600
- תש = (400+300) 700
- תת = (400+400) 800
- תררק = 400+200+200+100 = 900
Armenian numeral signs (minuscule letters):
units ա բ գ դ ե զ է ը թ 1 2 3 4 5 6 7 8 9 tens ժ ի լ խ ծ կ հ ձ ղ 10 20 30 40 50 60 70 80 90 hundreds ճ մ յ ն շ ո չ պ ջ 100 200 300 400 500 600 700 800 900 thousands ռ ս վ տ ր ց ւ փ ք 1000 2000 3000 4000 5000 6000 7000 8000 9000 ten-thousands օ ֆ 10000 20000 346 = յխզ
Unlike many alphabetic numeral systems, the Armenian system does not use multiplication by 1,000 or 10,000 in order to express higher values. Instead, higher values were written out in full using lexical numerals.[8]
Higher numbers
As the alphabet ended, various multiplicative methods were used for the expression of higher numbers in the different systems. In the Greek alphabetic system, for multiples of 1,000, the hasta sign was placed to the left below a numeral-sign to indicate that it should be multiplied by 1,000.[9]
- β = 2
- ͵β = 2,000
- ͵κ = 20,000
With a second level of multiplicative method – multiplication by 10,000 – the numeral set could be expanded. The most common method, used by Aristarchus, involved placing a numeral-phrase above a large M character (M = myriads = 10,000) to indicate multiplication by 10,000.[10] This method could express numbers up to 100,000,000 (108).
20,704 − (2 ⋅ 10,000 + 700 + 4) could be represented as:
ψδ = 20,704
According to Pappus of Alexandria's report, Apollonius of Perga used another method. In it, the numerals above M = myriads = 10,000 represented the exponent of 10,000. The number to be multiplied by M was written after the M character.[11] This method could express 5,462,360,064,000,000 as:
͵EYZB ͵ΓX ͵FY 100003 × 5462 + 100002 × 3600 + 100001 × 6400
Distinguishing numeral-phrases from text
Alphabetic numerals were distinguished from the words with special signs, most commonly a horizontal stroke above the numeral-phrase, but occasionally with dots placed to either side of it. The latter was manifested in the Greek alphabet with the hasta sign.
= 285
In Ethiopic numerals, known as Geʽez, the signs have marks both above and below them to indicate that their value is numerical. The Ethiopic numerals are the exception, where numeral signs are not letters of their script. This practice became universal from the 15th century onwards.[12]
Numeral signs of Ethiopic numerals with marks both above and below the letters:
1 2 3 4 5 6 7 8 9 × 1 ፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱ × 10 ፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ × 100 ፻ × 10,000 ፼
The direction of numerals follows the writing system's direction. Writing is from left to right in Greek, Coptic, Ethiopic, Ghotic, Armenian, Georgian, Glagolitic, and Cyrillic alphabetic numerals along with Shirakatsi's notation. Right-to-left writing is found in Hebrew and Syriac alphabetic numerals, Arabic abjad numerals, and Fez numerals.[citation needed]
Fractions
Unit fractions
Unit fractions were a method to express fractions. In Greek alphabetic notation, unit fractions were indicated with the denominator – alphabetic numeral sign – followed by small accents or strokes placed to the right of a numeral, known as a keraia (ʹ). Therefore, γʹ indicated one third, δʹ one fourth, and so on. These fractions were additive and were also known as Egyptian fractions.
For example: δ´ ϛ´ = 1⁄4 + 1⁄6 = 5⁄12.
A mixed number could be written as such: ͵θϡϟϛ δ´ ϛ´ = 9996 + 1⁄4 + 1⁄6
Astronomical fractions
In many astronomical texts, a distinct set of alphabetic numeral systems blend their ordinary alphabetical numerals with a base of 60, such as
With this
Astronomical fractions (with Greek alphabetic signs):
units α β γ δ ε ϛ ζ η θ 1 2 3 4 5 6 7 8 9 tens ι κ λ μ ν 10 20 30 40 50
͵αφιε κ ιε = 1515 + (20 x 1/60) + (15 x 1/3600) = 1515.3375
This blended system did not use a
This sexagesimal notation was especially useful in astronomy and mathematics because of the division of the circle into 360 degrees (with subdivisions of 60 minutes per degree and 60 seconds per minute). In Theon of Alexandria's (4th century AD) commentary on the Almagest, the numeral-phrase ͵αφιε κ ιε expresses 1515 (͵αφιε) degrees, 20 (κ) minutes, and 15 (ιε) seconds.[17] The degree's value is in the ordinary decimal alphabetic numerals, including the use of the multiplicative hasta for 1000, while the latter two positions are written in sexagesimal fractions.
Arabs adopted astronomical fractions directly from the Greeks, and similarly Hebrew astronomers used sexagesimal fractions, but Greek numeral signs were replaced by their own alphabetic numeral signs to express both integers and fractions.
Alphabetic numeral systems
See also
References
Citations
- ^ a b c d e Chrisomalis (2010), p. 185.
- ^ Chrisomalis (2010), p. 135–138.
- ^ Chrisomalis (2010), p. 155.
- ^ Chrisomalis (2010), p. 148.
- ^ Chrisomalis (2010), p. 152.
- ^ Chrisomalis (2010), p. 166.
- ^ Chrisomalis (2010), p. 156.
- ^ Chrisomalis (2010), p. 174.
- ^ Chrisomalis (2010), p. 138.
- ^ Heath (1921), pp. 39–41.
- ^ Greek number systems – MacTutor
- ^ Ifrah (1998), pp. 246–247.
- ^ Ifrah (1998), p. 156.
- ^ Chrisomalis (2010), p. 169.
- ^ Heath (1921), pp. 44–45.
- ^ Irani (1955).
- ^ Thomas (1962), pp. 50–51.
Bibliography
- Chrisomalis, Stephen (2010). Numerical Notation: A Comparative History. Cambridge University Press. pp. 133–187. ISBN 9780521878180.
- Heath, Thomas L. (1921). A History of Greek Mathematics. 2 vols. Cambridge: Cambridge University Press.
- Ifrah, Georges (1998). The universal history of numbers: from prehistory to the invention of the computer. Translated by Bellos, David. London: Harvill Press. ISBN 9781860463242.
- Irani, Rida A. K. (1955). "Arabic numeral forms". Centaurus. 4 (1): 1–12. .
- Thomas, Ivor (1962). Selections Illustrating the History of Greek Mathematics. Vol. 1. Cambridge, MA: Harvard University Press.
Further reading
- Bender, Marvin L., Sydney W. Head, and Roger Cowley. 1976. The Ethiopian writing system. In Language in Ethiopia, M.L. Bender, J.D. Bowen, R.L. Cooper, and CA. Ferguson, eds., pp. 120–129. London: Oxford University Press.
- Braune, Wilhelm and Ernst Ebbinghaus. 1966. Gotische Grammatik. Tübingen: Max Niemeyer Verlag.
- Colin, G.S. 1960. Abdjad. In Encyclopedia of Islam, vol. 1, pp. 97–98. Leiden: Brill.
- Colin, G.S. 1971. Hisab al-djummal. In Encyclopedia of Islam, vol. 3, p. 468. Leiden: Brill.
- Cubberley, Paul. 1996. The Slavic alphabets. In The World's Writing Systems, Peter T. Daniels and William Bright, eds., pp. 346–355. New York: Oxford University Press.
- Gandz, Solomon. 1933. Hebrew numerals. Proceedings of the American Academy of Jewish Research 4: pp. 53–112.
- Millard, A. 1995. Strangers from Egypt and Greece – the signs for numbers in early Hebrew. In Immigration and Emigration within the Ancient Near East, K. van Lerberghe and A. Schoors, eds., pp. 189–194. Leuven: Peeters.
- Megally, Fuad (1991). "Numerical system, Coptic". In Atiya, Azis S. (ed.). Coptic Encyclopedia. New York: Macmillan. pp. 1820–1822.
- Messiha, Heshmat. 1994. Les chiffres coptes. Le Monde Copte 24: 25–28.
- Neugebauer, Otto (1979). Ethiopic Astronomy and Computus. Wien: Verlag der Österreichischen Akademie der Wissenschaften.
- Pankhurst, Richard K. P., ed. 1985. Letters from Ethiopian Rulers (Early and Mid-Nineteenth Century), translated by David L. Appleyard and A.K. Irvine. Oxford: Oxford University Press.
- Schanzlin, G.L. 1934. The abjad notation. The Moslem World 24: 257–261.
- Shaw, Allen A. 1938–9. An overlooked numeral system of antiquity. National Mathematics Magazine 13: 368–372.
- Smith, David E. and L. C Karpinski. 1911. The Hindu-Arabic Numerals. Boston: Ginn