# Download PDF by A. Borel, G. Mostow: Algebraic Groups and Discontinuous Subgroups

By A. Borel, G. Mostow

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It is not in the course of solving problems that al-Khw¯arizm¯ı finds these equations. The classification in fact precedes the problems. 5. History 43 and second degree; and this theory will become the nucleus of a mathematical discipline. 2. This equation is not distinguished in any mathematical sense, but since it is the first one for which al-Khwarizmi employs the technique of completing the square, it has resonated through history. Or, as Karpinski suggests [36, p. ” Here is al-Khwarizmi’s own description of the example [54, p.

As we know, this reduces to solving a quadratic equation. Jumping ahead five centuries to the late stages of the Greek tradition, we come to Diophantus, another Alexandrian. Diophantus wrote Arithmetica, a collection of about 200 problems and their solutions. Arithmetica consisted of thirteen chapters, of which six were preserved in Greek and four in an Arabic translation from the ninth century, but the Arabic chapters were essentially lost until their rediscovery in 1968. In contrast to Euclid’s geometric algebra, Diophantus works in a manner we would recognize as more strictly algebraic, with symbols for the lower ✐ ✐ ✐ ✐ ✐ ✐ “IrvingBook” — 2013/5/22 — 15:39 — page 40 — #56 ✐ ✐ 40 2.

Rashed also writes about the traditions of calculation in the eighth century and alKhwarizmi’s knowledge of Greek and Indian mathematical literature. At the outset of his book, al-Khwarizmi introduces three types of quantities: roots, squares, and numbers, the root being what we would label x, the square being x 2 , and number being number. He then gets right to business, classifying linear and quadratic equations into six forms and illustrating how to solve each with examples. We would regard all six types as one, but since al-Khwarizmi does not use negative numbers or 0, he is obliged to consider separate cases, with a, b, and c always positive: (1) Squares equal roots, or what we would describe as ax 2 D bx.

### Algebraic Groups and Discontinuous Subgroups by A. Borel, G. Mostow

by Thomas

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