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ISLAMIC MATHEMATICS
The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia and parts of India from the 8th Century onwards made significant contributions towards mathematics. They were able to draw on and fuse together the mathematical developments of both Greece and India. One consequence of the Islamic prohibition on depicting the human form was the extensive use of complex geometric patterns to decorate their buildings, raising mathematics to the form of an art. In fact, over time, Muslim artists discovered all the different forms of symmetry that can be depicted on a 2dimensional surface. The Qu’ran itself encouraged the accumulation of knowledge, and a Golden Age of Islamic science and mathematics flourished throughout the medieval period from the 9th to 15th Centuries. The House of Wisdom was set up in Baghdad around 810, and work started almost immediately on translating the major Greek and Indian mathematical and astronomy works into Arabic. The outstanding Persian mathematician Muhammad AlKhwarizmi was an early Director of the House of Wisdom in the 9th Century, and one of the greatest of early Muslim mathematicians. Perhaps AlKhwarizmi’s most important contribution to mathematics was his strong advocacy of the Hindu numerical system (1  9 and 0), which he recognized as having the power and efficiency needed to revolutionize Islamic (and, later, Western) mathematics, and which was soon adopted by the entire Islamic world, and later by Europe as well. AlKhwarizmi's other important contribution was algebra, and he introduced the fundamental algebraic methods of “reduction” and “balancing” and provided an exhaustive account of solving polynomial equations up to the second degree. In this way, he helped create the powerful abstract mathematical language still used across the world today, and allowed a much more general way of analyzing problems other than just the specific problems previously considered by the Indians and Chinese.
The 10th Century Persian mathematician Muhammad AlKaraji worked to extend algebra still further, freeing it from its geometrical heritage, and introduced the theory of algebraic calculus. AlKaraji was the first to use the method of proof by mathematical induction to prove his results, by proving that the first statement in an infinite sequence of statements is true, and then proving that, if any one statement in the sequence is true, then so is the next one. Among other things, AlKaraji used mathematical induction to prove the binomial theorem. A binomial is a simple type of algebraic expression which has just two terms which are operated on only by addition, subtraction, multiplication and positive wholenumber exponents, such as (x + y)^{2}. The coefficients needed when a binomial is expanded form a symmetrical triangle, usually referred to as Pascal’s Triangle after the 17th Century French mathematician Blaise Pascal, although many other mathematicians had studied it centuries before him in India, Persia, China and Italy, including AlKaraji. Some hundred years after AlKaraji, Omar Khayyam (perhaps better known as a poet and the writer of the “Rubaiyat”, but an important mathematician and astronomer in his own right) generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots in the early 12th Century. He carried out a systematic analysis of cubic problems, revealing there were actually several different sorts of cubic equations. Although he did in fact succeed in solving cubic equations, and although he is usually credited with identifying the foundations of algebraic geometry, he was held back from further advances by his inability to separate the algebra from the geometry, and a purely algebraic method for the solution of cubic equations had to wait another 500 years and the Italian mathematicians del Ferro and Tartaglia.
The 13th Century Persian astronomer, scientist and mathematician Nasir AlDin AlTusi was perhaps the first to treat trigonometry as a separate mathematical discipline, distinct from astronomy. Building on earlier work by Greek mathematicians such as Menelaus of Alexandria and Indian work on the sine function, he gave the first extensive exposition of spherical trigonometry, including listing the six distinct cases of a right triangle in spherical trigonometry. One of his major mathematical contributions was the formulation of the famous law of sines for plane triangles, ^{a}⁄_{(sin A)} = ^{b}⁄_{(sin B)} = ^{c}⁄_{(sin C)}, although the sine law for spherical triangles had been discovered earlier by the 10th Century Persians Abul Wafa Buzjani and Abu Nasr Mansur. Other medieval Muslim mathematicians worthy of note include: With the stifling influence of the Turkish Ottoman Empire from the 14th or 15th Century onwards, Islamic mathematics stagnated, and further developments moved to Europe.
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