Kushyar ibn Labban

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Abu-l-Hasan Kushayr ibn Labban ibn Bashahri al-Jili (971 - 1029), was a Persian mathematician, geographer, and astronomer from Jilan, a.k.a. Gilan, south of the Caspian Sea, Iran.

His main work was probably done about the beginning of the eleventh century, and seems to have taken an important part in the elaboration of trigonometry. For example, he continued the investigations of Abul Wáfa, and devoted much space to this in his tables, al-zij al-jami wa-l-baligh ("the comprehensive and mature tables"), which were translated into Persian before the end of the century. He wrote also an astrological introduction and an arithmetic treatise (extant to Hebrew).

Most significant is his work on Hindu reckoning being the earliest known work on Arabic arithmetic which deals with Hindu numerals. An earlier text by Abu'l-Wafa on arithmetic did not use Hindu numerals. Kushyar's Principles of Hindu reckoning was written about 1000 AD.

There, Kushyar uses a symbol for zero, but does not use any separating symbol to distinguish the fractional part of a number from the integral part. He discusses decimal numbers in the main body of the text, relegating sexagesimal numbers to a separate treatment in tables. Topics considered include addition and subtraction of decimal numbers followed by multiplication and division of decimal numbers. Kushyar gives methods to construct exact square roots, as well as approximate methods to calculate the square roots of non-square numbers. Similarly he gives methods to construct exact cube roots, and an approximate method to calculate the cube root of a non-square number. As a check on the accuracy of his results, Kushyar uses the method of "casting out nines" or "checking the nines" which basically checks that the sums are correct modulo 9.

He was the teacher of Ahmad Nasawi. He is thought to have died in Baghdad.

[edit] Sources

H. Suter: Mathematiker und Astronomen der Araber (83, 235, 1900; 168, 1902).
M Levey and M Petruck (trs.), Kushyar ibn Labban, Principles of Hindu reckoning (Madison, 1965).