We also have tables of squares, of cubes. Other tables contain approximate reciprocals of irregular numbers. There are multiplication tables headed by numbers that are the reciprocals c -1 of simple regular integers. The usual procedure to divide b by c was to take c -1 from a table of reciprocals and to calculate b c -1by means of a multiplication table. Such integers may be called “regular.” A normal table of reciprocals would give the reciprocals of all regular integers from 2 to 1,21 as follows:įrom the Hellenistic era we have large tables giving the reciprocals of regular numbers up to seven sexagesimal places (see Neugebauer, MKT, 1, 16). The reciprocal of an integer is a finite sexagesimal fraction, provided the integer contains no prime factors other than 2, 3, and 5. the table would contain the itemsĪll other multiples of 1,30 were obtained by adding items found in the table. The Sumerians already had multiplication tables, each table containing the multiples of a given “head number,” For instance, if the head number is 1,30.
The Sumerians and Babylonians performed such arithmetical operations as multiplication, division, squaring, and extracting square and cube roots by means of tables. Thureau-Dangin, Esquisse d’une histoire du système sexagésimal (Paris. Neugebauer, “Zur Entstehung des Sexagesimalsystems.” in Abhandlungen der Akademie der Wissenschaften zu Göttingen, n.s. On the other hand, if it is known that the first unit has the value 60 2. If it is known that the first unit has the value 1, integers and fractions will he separated by a semicolon: Neugebauer, separating successive sexagesimal places by a comma: In transcribing Sumerian or Babylonian numerals, I shall follow the method introduced by O. together with cuneiform script, which they adapted to their Semitic languages. the Babylonian and Assyrian scribes inherited this system from the Sumerians. Thus, 1 1/2 would be written as 1,30 and the reader would have to determine from the context whether I 1/2 or 90 was meant, or even 90 X 60 n or 1 1/2 X 60 -n.Īfter 2000 B.C. The denominator 60 n was not written: it had to be inferred from the context. The normal way of writing fractions was to express them as multiples of 1/60, or of 1/60 2, and so on. Beyond 59, integers were written as sums of multiples of powers of 60. , the symbols for 10 and 1 being repeated as often as necessary. Integers up to 59 were written in the decimal system. The symbol for I, a vertical wedge, can denote any power of 60 or any power of 1/60. “Positional” means that the value of a numeral depends on where it stands. the inventors of cuneiform script, who lived in southern Mesopotamia, used a positional system with base 60 for writing integers and fractions. The Sexagesimal System, Before 2000 B.C.
Textes mathématiques babyloniens (Leiden, 1938). Sachs, Mathematical Cuneiform Texts, which is American Oriental Series. Neugebauer, Mathematische Keilschrifttexte, 3 vols., which is Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt.
BABYLONIAN NUMERALS 83 SERIES
Rutten, Textes mathématiques de Suse, in the series Mémoires de la Mission Archéologique en Iran (Paris. Babylonian mathematics can be studied without any attention to astronomy. Mathematical astronomy came into existence much later. ).īabylonian arithmetic, algebra, and geometry were already fully developed under the dynasty of Hammurapi. probably 1830–1531 B.C.) or Hellenistic (Seleucid era, beginning 311 B.C. The majority of the texts are either Old Babylonian (dynasty of Hammurapi. Rutten produced a volume entitled Textes mathématiques de Suse (1960, in which cuneiform texts from Susa were published and discussed.Īll our knowledge concerning Babylonian mathematics is ultimately derived from these sources. It contains newly discovered texts and corrections to earlier publications. Mathematical Cuneiform Texts (MCT), supplementing Neugebauer’s MKT. In 1945, Neugebauer and Sachs published another collection. Thureau-Dangin contains new transcriptions and French translations of most texts published in Neugebauer’s MKT, with a new, valuable commentary. They contain photographs, transcriptions, and German translations of all texts, with a very good commentary.
In 1935-1937, Otto Neugebauer published all mathematical cuneiform texts then known in three volumes entitled Mathematische Keilschriftterte (MKT).
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