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Al-Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, which was syncopated, meaning that some symbolism is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of Ibn al-Banna' al-Marrakushi and Ab? al-?asan ibn ?Al? al-Qalad?.
"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for the future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before."
Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. Omar Khayyam, along with Sharaf al-D?n al-T?s?, found several solutions of the cubic equation. Omar Khayyam found the general geometric solution of a cubic equation.
To solve the third-degree equation x3 + a2x = b Khayyám constructed the parabolax2 = ay, a circle with diameter b/a2, and a vertical line through the intersection point. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis.
Omar Khayyam (c. 1038/48 in Iran - 1123/24) wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of cubic or third-order equations, going beyond the Algebra of al-Khw?rizm?. Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections. This method had been used by the Greeks, but they did not generalize the method to cover all equations with positive roots.
Sharaf al-D?n al-s? (? in Tus, Iran - 1213/4) developed a novel approach to the investigation of cubic equations--an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. For example, to solve the equation , with a and b positive, he would note that the maximum point of the curve occurs at , and that the equation would have no solutions, one solution or two solutions, depending on whether the height of the curve at that point was less than, equal to, or greater than a. His surviving works give no indication of how he discovered his formulae for the maxima of these curves. Various conjectures have been proposed to account for his discovery of them.
The Greeks had discovered irrational numbers, but were not happy with them and only able to cope by drawing a distinction between magnitude and number. In the Greek view, magnitudes varied continuously and could be used for entities such as line segments, whereas numbers were discrete. Hence, irrationals could only be handled geometrically; and indeed Greek mathematics was mainly geometrical. Islamic mathematicians including Ab? K?mil Shuj ibn Aslam and Ibn Tahir al-Baghdadi slowly removed the distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations. They worked freely with irrationals as mathematical objects, but they did not examine closely their nature.
^Katz (1993): "A complete history of mathematics of medieval Islam cannot yet be written, since so many of these Arabic manuscripts lie unstudied... Still, the general outline... is known. In particular, Islamic mathematicians fully developed the decimal place-value number system to include decimal fractions, systematised the study of algebra and began to consider the relationship between algebra and geometry, studied and made advances on the major Greek geometrical treatises of Euclid, Archimedes, and Apollonius, and made significant improvements in plane and spherical geometry."
Smith (1958) Vol. 1, Chapter VII.4: "In a general way it may be said that the Golden Age of Arabian mathematics was confined largely to the 9th and 10th centuries; that the world owes a great debt to Arab scholars for preserving and transmitting to posterity the classics of Greek mathematics; and that their work was chiefly that of transmission, although they developed considerable originality in algebra and showed some genius in their work in trigonometry."
^Adolph P. YushkevichSertima, Ivan Van (1992), Golden age of the Moor, Volume 11, Transaction Publishers, p. 394, ISBN1-56000-581-5 "The Islamic mathematicians exercised a prolific influence on the development of science in Europe, enriched as much by their own discoveries as those they had inherited by the Greeks, the Indians, the Syrians, the Babylonians, etc."
^Berggren, J. Lennart; Al-T?s?, Sharaf Al-D?n; Rashed, Roshdi (1990). "Innovation and Tradition in Sharaf al-D?n al-s?'s al-Mudal?t". Journal of the American Oriental Society. 110 (2): 304-309. doi:10.2307/604533. JSTOR604533.
^ abSesiano, Jacques (2000). Helaine, Selin; Ubiratan, D'Ambrosio, eds. Islamic mathematics. Mathematics Across Cultures: The History of Non-western Mathematics. Springer. pp. 137-157. ISBN1-4020-0260-2.
Review: Hogendijk, Jan P.; Berggren, J. L. (1989). "Episodes in the Mathematics of Medieval Islam by J. Lennart Berggren". Journal of the American Oriental Society. American Oriental Society. 109 (4): 697-698. doi:10.2307/604119. JSTOR604119.
Youschkevitch, Adolf P.; Rozenfeld, Boris A. (1960). Die Mathematik der Länder des Ostens im Mittelalter. Berlin. Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62-160.
Youschkevitch, Adolf P. (1976). Les mathématiques arabes: VIIIe-XVe siècles. translated by M. Cazenave and K. Jaouiche. Paris: Vrin. ISBN978-2-7116-0734-1.
Book chapters on Islamic mathematics
Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Victor J. Katz. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Second ed.). Princeton, New Jersey: Princeton University. ISBN978-0-691-11485-9.