Mathematics in medieval Islam
The history of mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, building on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta), saw important developments, such as the full development of the decimal placevalue system to include decimal fractions, the first systematised study of algebra (named for the work of scholar AlKwarizmi), and advances in geometry and trigonometry.^{[1]} Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries.^{[2]}
Contents
History
Algebra
The study of algebra, which itself is an Arabic word meaning "reunion of broken parts",^{[3]} flourished during the Islamic golden age. AlKhwarizmi is, along with the Greek mathematician Diophantus, known as the father of algebra.^{[4]} In his book The Compendious Book on Calculation by Completion and Balancing AlKhwarizmi deals with ways to solve for the positive roots of first and second degree (linear and quadratic) polynomial equations.^{[5]} He also introduces the method of reduction, and unlike Diophantus, gives general solutions for the equations he deals with.^{[4]}
AlKhwarizmi'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, where some symbolism is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of Ibn alBanna' alMarrakushi and Abū alḤasan ibn ʿAlī alQalaṣādī.^{[6]}
On the work done by AlKhwarizmi, J. J. O'Connor and Edmund F. Robertson said:^{[7]}
"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of alKhwarizmi, 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 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 AlKhwarizmi. Omar Khayyam, along with Sharaf alDīn alTūsī, found several solutions of the cubic equation. Omar Khayyam found the general geometric solution of a cubic equation.
Irrational numbers
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 Abū Kāmil Shujāʿ ibn Aslam 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 objects, but they did not examine closely their nature.^{[8]}
In the twelfth century, Latin translations of AlKhwarizmi's Arithmetic on the Indian numerals introduced the decimal positional number system to the Western world.^{[9]} His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations. In Renaissance Europe, he was considered the original inventor of algebra, although it is now known that his work is based on older Indian or Greek sources.^{[10]} He revised Ptolemy's Geography and wrote on astronomy and astrology.
Induction
The earliest implicit traces of mathematical induction can be found in Euclid's proof that the number of primes is infinite (c. 300 BCE). The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique (1665).
In between, implicit proof by induction for arithmetic sequences was introduced by alKaraji (c. 1000) and continued by alSamaw'al, who used it for special cases of the binomial theorem and properties of Pascal's triangle.
Major figures and developments
Omar Khayyám
Omar Khayyám (c. 1038/48 in Iran – 1123/24)^{[11]} wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of thirddegree equations, going beyond the Algebra of Khwārazmī.^{[12]} Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections. This method had been used by the Greeks,^{[13]} but they did not generalize the method to cover all equations with positive roots.^{[14]}
Sharaf alDīn alṬūsī
Sharaf alDīn alṬūsī (? 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.^{[15]}
Other major figures
 'Abd alHamīd ibn Turk (fl. 830) (quadratics)
 Thabit ibn Qurra (826–901)
 Abū Kāmil Shujā ibn Aslam (c. 850 – 930) (irrationals)
 Sind ibn Ali
 Abū Sahl alQūhī (c. 940–1000) (centers of gravity)
 Abu'lHasan alUqlidisi (952 – 953) (arithmetic)
 'Abd al'Aziz alQabisi
 Abū alWafā' Būzjānī (940 – 998) (spherical trigonometry)
 AlKaraji (c. 953 – c. 1029) (algebra, induction)
 Abu Nasr Mansur (c. 960 – 1036) (spherical trigonometry)
 Ibn Tahir alBaghdadi (c. 980–1037) (irrationals)
 Ibn alHaytham (ca. 965–1040)
 Abū alRayḥān alBīrūnī (973 – 1048) (trigonometry)
 Ibn Yaḥyā alMaghribī alSamawʾal (c. 1130 – c. 1180)
 Ibn Maḍāʾ (c. 1116  1196)
 Naṣīr alDīn alṬūsī (1201–1274) (parallel postulate)
 Jamshīd alKāshī (c. 1380–1429) (decimals and estimation of the circle constant)
Gallery

Engraving of Abū Sahl alQūhī's perfect compass to draw conic sections.
See also
 Arabic numerals
 Islamic Golden Age
 Timeline of Islamic science and technology
 Indian influence on Islamic mathematics in medieval Islam
 History of algebra
 History of calculus
 History of geometry
 History of trigonometry
Notes
 ↑ 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 placevalue 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. Yushkevich Sertima, Ivan Van (1992), Golden age of the Moor, Volume 11, Transaction Publishers, p. 394, ISBN 1560005815<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> "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."
 ↑ "algebra". Online Etymology Dictionary.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ ^{4.0} ^{4.1} http://fabpedigree.com/james/mathmen.htm#Khowarizmi
 ↑ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. p. 228. ISBN 0471543977.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ O'Connor, John J.; Robertson, Edmund F., "alMarrakushi ibn AlBanna", MacTutor History of Mathematics archive, University of St Andrews<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ http://www.math.tamu.edu/~dallen/history/infinity.pdf
 ↑ Struik 1987, p. 93
 ↑ Rosen 1831, p. v–vi; Toomer 1990
 ↑ Struik 1987, p. 96.
 ↑ Boyer 1991, pp. 241–242.
 ↑ Struik 1987, p. 97.
 ↑ Boyer 19991, pp. 241–242.
 ↑ Berggren, J. Lennart; AlTūsī, Sharaf AlDīn; Rashed, Roshdi; AlTusi, Sharaf AlDin (1990), "Innovation and Tradition in Sharaf alDīn alṬūsī's alMuʿādalāt", Journal of the American Oriental Society, 110 (2): 304–309, doi:10.2307/604533, JSTOR 604533<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
References
 Boyer, Carl B. (1991), "Greek Trigonometry and Mensuration, and The Arabic Hegemony", A History of Mathematics (2nd ed.), New York City: John Wiley & Sons, ISBN 0471543977CS1 maint: ref=harv (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Katz, Victor J. (1993), A History of Mathematics: An Introduction, HarperCollins college publishers, ISBN 0673380394CS1 maint: ref=harv (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 Ronan, Colin A. (1983), The Cambridge Illustrated History of the World's Science, Cambridge University Press, ISBN 0521258448CS1 maint: ref=harv (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Smith, David E. (1958), History of Mathematics, Dover Publications, ISBN 0486204294CS1 maint: ref=harv (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Struik, Dirk J. (1987), A Concise History of Mathematics (4th rev. ed.), Dover Publications, ISBN 0486602559CS1 maint: ref=harv (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
Further reading
 Books on Islamic mathematics
 Berggren, J. Lennart (1986), Episodes in the Mathematics of Medieval Islam, New York: SpringerVerlag, ISBN 0387963189<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Review: Toomer, Gerald J.; Berggren, J. L. (1988), "Episodes in the Mathematics of Medieval Islam", American Mathematical Monthly, Mathematical Association of America, 95 (6): 567, doi:10.2307/2322777, JSTOR 2322777<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 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, JSTOR 604119<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>)
 Daffa', Ali Abdullah al (1977), The Muslim contribution to mathematics, London: Croom Helm, ISBN 0856644641<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Rashed, Roshdi (2001), The Development of Arabic Mathematics: Between Arithmetic and Algebra, Transl. by A. F. W. Armstrong, Springer, ISBN 0792325656<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Youschkevitch, Adolf P. (1960), Die Mathematik der Länder des Ostens im Mittelalter, Berlin Unknown parameter
coauthors=
ignored (help)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62–160.  Youschkevitch, Adolf P. (1976), Les mathématiques arabes: VIII^{e}–XV^{e} siècles, translated by M. Cazenave and K. Jaouiche, Paris: Vrin, ISBN 9782711607341<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Book chapters on Islamic mathematics
 Berggren, J. Lennart (2007), "Mathematics in Medieval Islam", in Victor J. Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Second ed.), Princeton, New Jersey: Princeton University, ISBN 9780691114859<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Cooke, Roger (1997), "Islamic Mathematics", The History of Mathematics: A Brief Course, WileyInterscience, ISBN 0471180823<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Books on Islamic science
 Daffa, Ali Abdullah al; Stroyls, J.J. (1984), Studies in the exact sciences in medieval Islam, New York: Wiley, ISBN 0471903205<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Kennedy, E. S. (1984), Studies in the Islamic Exact Sciences, Syracuse Univ Press, ISBN 0815660677<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Books on the history of mathematics
 Joseph, George Gheverghese (2000), The Crest of the Peacock: NonEuropean Roots of Mathematics (2nd ed.), Princeton University Press, ISBN 0691006598<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> (Reviewed: Katz, Victor J.; Joseph, George Gheverghese (1992), "The Crest of the Peacock: NonEuropean Roots of Mathematics by George Gheverghese Joseph", The College Mathematics Journal, Mathematical Association of America, 23 (1): 82–84, doi:10.2307/2686206, JSTOR 2686206<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>)
 Youschkevitch, Adolf P. (1964), Gesichte der Mathematik im Mittelalter, Leipzig: BG Teubner Verlagsgesellschaft<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Journal articles on Islamic mathematics
 Høyrup, Jens. “The Formation of «Islamic Mathematics»: Sources and Conditions”. Filosofi og Videnskabsteori på Roskilde Universitetscenter. 3. Række: Preprints og Reprints 1987 Nr. 1.
 Bibliographies and biographies
 Brockelmann, Carl. Geschichte der Arabischen Litteratur. 1.–2. Band, 1.–3. Supplementband. Berlin: Emil Fischer, 1898, 1902; Leiden: Brill, 1937, 1938, 1942.
 Sánchez Pérez, José A. (1921), Biografías de Matemáticos Árabes que florecieron en España, Madrid: Estanislao Maestre<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Sezgin, Fuat (1997), Geschichte Des Arabischen Schrifttums (in German), Brill Academic Publishers, ISBN 9004020071CS1 maint: unrecognized language (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Suter, Heinrich (1900), Die Mathematiker und Astronomen der Araber und ihre Werke, Abhandlungen zur Geschichte der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen, X Heft, Leipzig<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Television documentaries
 Marcus du Sautoy (presenter) (2008). "The Genius of the East". The Story of Maths. BBC.
 Jim AlKhalili (presenter) (2010). Science and Islam. BBC.
External links
 Hogendijk, Jan P. (January 1999). "Bibliography of Mathematics in Medieval Islamic Civilization".CS1 maint: ref=harv (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 Richard Covington, Rediscovering Arabic Science, 2007, Saudi Aramco World