page_title: Alphabetic numeral system
section_title: Astronomical fractions
hierarchical_section_title: Alphabetic numeral system / Fractions / Astronomical fractions
caption_attribution_description: English: Fragment of a Greek papyrus from the 2nd century C.E., showing an early version of the zero sign (lower right corner). This papyrus is housed at Lund University Library and has the inventory ID: P. Lund. Inv. 35.
context_page_description: An alphabetic numeral system is a type of numeral system. Developed in classical antiquity, it flourished during the early Middle Ages. In alphabetic numeral systems, numbers are written using the characters of an alphabet, syllabary, or another writing system. Unlike acrophonic numeral systems, where a numeral is represented by the first letter of the lexical name of the numeral, alphabetic numeral systems can arbitrarily assign letters to numerical values. Some systems, including the Arabic, Georgian and Hebrew systems, use an already established alphabetical order. Alphabetic numeral systems originated with Greek numerals around 600 BC and became largely extinct by the 16th century. After the development of positional numeral systems like Hindu–Arabic numerals, the use of alphabetic numeral systems dwindled to predominantly ordered lists, pagination, religious functions, and divinatory magic.
context_section_description: In many astronomical texts, a distinct set of alphabetic numeral systems blend their ordinary alphabetical numerals with a base of 60, such as Babylonian sexagesimal systems. In the 2nd century BC, a hybrid of Babylonian notation and Greek alphabetic numerals emerged and was used to express fractions. Unlike the Babylonian system, the Greek base of 60 was not used for expressing integers.   
  With this sexagesimal positional system – with a subbase of 10 – for expressing fractions, fourteen of the alphabetic numerals were used (the units from 1 to 9 and the decades from 10 to 50) in order to write any number from 1 through 59. These could be a numerator of a fraction. The positional principle was used for the denominator of a fraction, which was written with an exponent of 60 (60, 3,600, 216,000, etc.). Sexagesimal fractions could be used to express any fractional value, with the successive positions representing 1/60, 1/60², 1/60³, and so on.  The first major text in which this blended system appeared was Ptolemy's Almagest, written in the 2nd century AD. 
  Astronomical fractions (with Greek alphabetic signs):
  This blended system did not use a radix point, but the astronomical fractions had a special sign to indicate zero as a placeholder. Some late Babylonian texts used a similar placeholder. The Greeks adopted this technique using their own sign, whose form and character changed over time from early manuscripts (1st century AD) to an alphabetic notation.
  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. 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.
caption: Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
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