Abu Ja'far Muhammad ibn Musa Al-Khwarizmi Biography

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World of Mathematics on Abu Ja'far Muhammad ibn Musa Al-Khwarizmi

The Arab astronomer Abu Ja'far Muhammad ibn Musa Al-Khwarizmi was the author of about a half dozen astronomical works, including a book entitled Al-jabr w'al muqabala (written in 830 AD) that gave the name al-jabr to the branch of mathematics that is now known by its modern spelling as algebra.

The word al-jabr is usually translated as "restoring," with reference to restoring the balance in an equation by placing on one side of an equation a term that has been removed from the other. For example, given the equation x²-5=4, a balance is restored by writing x²=9. The second part of the title, al muqabala, probably meant "simplification," as in the case of combining 2x+5x to obtain 7x, or by subtracting out equivalent terms from both sides of an equation.

Al-Khwarizmi's algebra was based on the earlier work of the Hindu mathematician Brahmagupta (b. 598), but also contained influences from Babylonia and Greek mathematics. Some of the operations that Al-Khwarzimi describes are the same as those described by the Greek mathematician Diophantus (c. 250 AD), although it is unlikely that Al-Khwarzimi was familiar with the latter's work. In the case of an equation is several unknowns, for example, both Diophantus and Al-Kwarzimi reduce the equation to one unknown and then solve it. Some of Al-Khwarzimi's terminology clearly reflects Diophantus' nomenclature; for example, both mathematicians use the term "power" to describe the square of an unknown. [Describing the powers of an unknown, Al-Khwarzimi designated the unknown as the root (as of a plant), which is the origin of our use of term.]

In the book, Al-Kwarizimi identifies the product of (x +/- a) and (y +/- b). Given an expression of the form ax² + bx +c, he describes how to add and subtract terms. When treating linear and quadratic equations, he followed Diophantus in retaining six separate forms in his solutions, i.e., ax² = bx, ax² = c, ax² + c = bx, ax² + bx =c, and ax2 = bx + c, where a, b and c were always positive. In this way, Al-Kwarizimi was able to avoid the problem of subtracting a larger number from a smaller one. Although Al-Khwarzimi recognized that quadratic equations could have two roots, he only listed the real positive ones (which could, of course, be irrational).

When Al-Khwarzimi wanted to describe the solution of a quadratic equation, he, like other Arab mathematicians, resorted to using geometrical constructions and completing the square.

It of interest to linguists that the term al-jabr later came to mean "bonesetter," i.e., the person who restores broken bones. After the term reached Spain with the Moorish invasion, it became algebrista, and continued to be a term for one who sets bones. At one time, it was common for Spanish barbers to post a sign outside their establishments reading "Algebrista y Sangrador" (meaning bonesetter and bloodletter), it being the custom for barbers to practice those medical arts. And the tradition of barbers performing these treatments continued for some centuries. In sixteenth-century Italy, the term algebra continued to be used for the practice of bonesetting.

When Al-Khwarizmi's book was translated into Latin in the twelfth century, the title was translated Ludus algebrae et almucgrabalaeque. The name for the branch of mathematics, however, was eventually shortened to algebra.

Another of Al-Khwarizimi's books on arithmetic and algebra, De numero indorum (Concerning the Hindu Art of Reckoning) has survived only in its Latin translation. In it, the author gave such a complete account of the Hindu numeral system that that system of numbering is now mistakenly known as Arabic numerals. When Latin translations began spreading throughout Europe, readers began crediting the mathematical notation, which became known as algorismi, to Al-Khwarzimi. Later, the scheme that made use of the Hindu numerals became known as algorithm (a corruption of al-Khwarzimi). Today, of course, the word algorithm refers to a set of well-defined rules for solving a problem in a finite number of steps.