KARAJI, AL-

born c. 980, most likely Karaj, Persia, rather than Karkh, near Baghdad, Iraq died c. 1030 also known as al-Karkhi, in full, Abu Bakr ibn Muhammad ibn al-Husayn al-Karaji mathematician and engineer who held an official position in Baghdad (c. 10101015), perhaps culminating in the position of vizier, during which time he wrote his three main works, al-Fakhri fi'l-jabr wa'l-muqabala (Glorious on algebra), al-Badi fi'l-hisab (Wonderful on calculation), and al-Kafi fi'l-hisab (Sufficient on calculation). A now lost work of his contained the first description of what later became known as Pascal's triangle (see binomial theorem). Al-Karaji combined tradition and novelty in his mathematical exposition. Like his Arabic predecessors he did not use symbolismeven writing numbers as words rather than using Indian numerals (except for large numbers and in numerical tables). However, with his writings Arabic algebra began to free itself from the early tradition of illustrating formulas and the resolutions of equations with geometric diagrams. As part of his official duties al-Karaji composed his Sufficient, an arithmetic textbook for civil servants on calculating with integers and fractions (in both base 10 and base 60), extracting square roots, and determining areas and volumes. He also composed a small and very elementary compendium of basic algebra. The Glorious and the Wonderful are more advanced algebraic texts and contain a large collection of problems. In particular, the Wonderful contains a useful introduction to the basic algebraic methods of Diophantus of Alexandria (fl. c. 250). Although much of his work was taken from others' writings, there is no doubt that al-Karaji was an able mathematician, and traces of his influence were frequent in the following centuries. However, the quality of his work was uneven; he seems to have worked too hastily at times, as he confessed in the closing words of the Sufficient. After leaving Baghdad for Persia, al-Karaji wrote an engineering work on drilling wells and building aqueducts. Additional reading Three excellent studies on al-Karaji 's main works are: Franz Woepcke, Extrait du Fakhr (1853, reprinted 1982); Adel Anbouba (ed.), L'Algbre al-badi d'al-Karagi (1964); and Adolf Hochheim (trans.), Kf fl Hisb (Gengendes ber Arithmetik), 3 parts (1878-80). Jacques Sesiano