In the year that William the Conqueror crossed the English Channel and conquered England, Abu al-Fath Omar ben Ibrahim al-Khayyam turned eighteen. Commonly known as Omar Khayyam, he is most famous for the quatrains from the Rubaiyat for which Edward FitzGerald has credited him as composing. However, Khayyam was also an "outstanding mathematician and astronomer." He calculated the length of the year to an extraordinary degree of accuracy, and his work in mathematics touched upon algebra, geometry, and number theory.
Khayyam finished writing On Demonstrations of Problems of Algebra and Almucabala in, or around, the year 1079. In this book he proposed a geometric means for solving cubic equations. His work was a natural extension of Euclid's method of solving quadratic equations, but, because of the additional degree, "Omar had to use more advanced methods, and so . . . he obtained his solutions by the intersection of conics." (See http://www-history.mcs.st-and.ac.uk:/history/Diagrams/KhayyamCubic.gif for an example.) In addition, researchers now believe that Omar was one of the earliest mathematicians to investigate the expansion of the binomial (a + b)n, because although Algebra does not include his work on the binomial expansion, Khayyam references one of his previous works that apparently does include his investigation of the binomial expansion.
Besides his work in algebra, Khayyam also examined the parallel axiom in another of his works, Commentaries on the Difficulties in the Postulates of Euclid's Elements. This axiom states "that through a point not on a line one and only one parallel can be drawn to this line . . . [and] appears (in a slightly different form) in Euclid's Elements as the Fifth Postulate." He attempts to prove the parallel axiom by constructing a rectangle and proving that the opposing line segments are parallel. His methodology to prove the parallel axiom required him to consider acute, obtuse, and right angles (although he did not name them as such at the time), and it appears that his work was the first investigating this matter.
Omar's book on Euclid also contains the rudimentary beginnings of the concept of real numbers. He discusses Euclid's definition of ratios and proposes his own definition for equal ratios. Khayyam defines two ratios as equal "when they can be expressed by the ratio of integer numbers with as great a degree of accuracy as we like." This definition of equivalent ratios suggests the existence of real numbers, whose definition later mathematicians established in the nineteenth century.
Author: Tony Brinsko
References:
D.J. Struik, Omar Khayyam, Mathematics Teacher 4 (1958), 280-286.
http://www-history.mcs.st-and.ac.uk:/history/Mathematicians/Khayyam.html
http://www.cwi.nl/~keesh/Iran/Maths/KHAYYAM.html
http://www-history.mcs.st-and.ac.uk:/history/Diagrams/KhayyamCubic.gif
http://www-leland.stanford.edu/~yuri/Omar/omar.html
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