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| [[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|right|A page from the ''[[The Compendious Book on Calculation by Completion and Balancing]]'' by [[Muhammad ibn Mūsā al-Khwārizmī|Al-Khwarizmi]].]]
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| {{pp-move-indef|small=yes}}In the [[history of mathematics]], '''mathematics in medieval Islam''', often called '''Islamic mathematics''' or '''Arabic mathematics''', covers the body of [[mathematics]] preserved and advanced under the [[Muslim world|Islamic civilization]] between circa 622 and c.1600.{{sfn|Hogendijk|1999}} [[Islamic science]] and mathematics flourished under the Islamic [[caliphate]] established across the Middle East, extending from the [[Iberian Peninsula]] in the west to the [[Indus]] in the east and to the [[Almoravid Dynasty]] and [[Mali Empire]] in the south.
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| In his ''A History of Mathematics'', Victor Katz says that:{{sfn|Katz|1993}}
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| <blockquote>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 of Perga|Apollonius]], and made significant improvements in plane and spherical geometry.</blockquote>
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| An important role was played by the translation and study of [[Greek mathematics]], which was the principal route of transmission of these texts to Western Europe. Smith notes:{{sfn|Smith|1958|loc=Vol. 1, Chapter VII.4}}
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| <blockquote>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.</blockquote>
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| [[Adolph P. Yushkevich]] states regarding the role of Islamic mathematics:<ref>{{Citation|last=Sertima|first=Ivan Van|title=Golden age of the Moor, Volume 11|year=1992|publisher=Transaction Publishers|isbn=1-56000-581-5|authorlink=Ivan van Sertima|page=394}}</ref>
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| <blockquote>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.</blockquote>
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| == History ==
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| [[File:Abu Reyhan Biruni-Earth Circumference.svg|200px|thumb|Al-Biruni developed a new method using trigonometric calculations to compute earth's [[radius]] and [[circumference]] based on the angle between the horizontal line and true horizon from the peak of a mountain with known height.<ref>{{MacTutor|id=Al-Biruni|title=Al-Biruni}}</ref><ref>{{Citation|last=Douglas|first=A. V.|title=R.A.S.C. Papers- Al-Biruni, Persian Scholar|journal=Journal of the Royal Astronomical Society of Canada|year=1973|volume=67|pages=973–1048|bibcode=1973JRASC..67..209D}}</ref>]]
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| The most important contribution of the Islamic mathematicians was the development of algebra; combining Indian and Babylonian material with the Greek geometry to develop algebra.
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| === Irrational numbers ===
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| The Greeks had discovered [[Irrational number]]s, 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.<ref>http://www.math.tamu.edu/~dallen/history/infinity.pdf</ref>
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| In the twelfth century, [[Latin]] translations of [[Al-Khwarizmi]]'s [[Muḥammad ibn Mūsā al-Khwārizmī#Arithmetic|Arithmetic]] on the [[Indian numerals]] introduced the [[decimal]] [[Positional notation|positional number system]] to the [[Western world]].<ref name="Struik 93">{{harvnb|Struik|1987| p= 93}}</ref> His ''[[Compendious Book on Calculation by Completion and Balancing]]'' presented the first systematic solution of [[linear equation|linear]] and [[quadratic equation]]s. 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.<ref>{{harvnb|Rosen|1831|p=v–vi}}; {{harvnb|Toomer|1990}}</ref> He revised [[Ptolemy]]'s ''[[Geography (Ptolemy)|Geography]]'' and wrote on astronomy and astrology.
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| === Induction ===
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| {{See also|Mathematical induction#History}}
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| The earliest implicit traces of mathematical induction can be found in [[Euclid]]'s [[Euclid's theorem|proof that the number of primes is infinite]] (c. 300 BCE). The first explicit formulation of the principle of induction was given by [[Blaise Pascal|Pascal]] in his ''Traité du triangle arithmétique'' (1665).
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| In between, implicit [[Mathematical proof|proof]] by induction for [[Arithmetic progression|arithmetic sequences]] was introduced by [[al-Karaji]] (c. 1000) and continued by [[Ibn Yahyā al-Maghribī al-Samaw'al|al-Samaw'al]], who used it for special cases of the [[binomial theorem]] and properties of [[Pascal's triangle]].
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| == Major figures and developments ==
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| === Omar Khayyám ===
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| [[File:Omar Kayyám - Geometric solution to cubic equation.svg|thumb|To solve the third-degree equation ''x''<sup>3</sup> + ''a''<sup>2</sup>''x'' = ''b'' Khayyám constructed the [[parabola]] ''x''<sup>2</sup> = ''ay'', a [[circle]] with diameter ''b''/''a''<sup>2</sup>, 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.]]
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| [[Omar Khayyám]] (c. 1038/48 in [[Iran]] – 1123/24){{sfn|Struik|1987|p=96}} wrote the ''Treatise on Demonstration of Problems of Algebra'' containing the systematic solution of [[third-degree equation]]s, going beyond the ''Algebra'' of [[Muḥammad ibn Mūsā al-Khwārizmī|Khwārazmī]].{{sfn|Boyer|1991|pp=241–242}} Khayyám obtained the solutions of these equations by finding the intersection points of two [[conic section]]s. This method had been used by the Greeks,{{sfn|Struik|1987|p=97}} but they did not generalize the method to cover all equations with positive [[Zero of a function|roots]].{{sfn|Boyer|19991|pp=241–242}}
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| Khayyám differentiated between "geometric" and "arithmetic" solutions.{{sfn|Struik|1987|p=97}} Khayyám mistakenly believed{{sfn|Boyer|1991|pp=241–242}} arithmetic solutions only existed if the [[root (equation)|roots]] where [[positive number|positive]] and [[rational number|rational]].{{sfn|Struik|1987|p=97}} Khayyám did not concern himself with numerical calculations of the solutions.{{sfn|Struik|1987|p=97}}
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| {{#tag:ref|"Omar Khayyam (ca. 1050–1123), the "tent-maker," wrote an ''Algebra'' that went beyond that of al-Khwarizmi to include equations of third degree. Like his predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). [...] For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, [...]"{{sfn|Boyer|1991|pp=241–242}}|group="note"}}
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| === Sharaf al-Dīn al-Ṭūsī ===
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| [[Sharaf al-Dīn al-Ṭūsī]] (? in [[Tus, Iran]] – 1213/4) developed a novel approach to the investigation of [[Cubic function|cubic equations]]—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. For example, to solve the equation <math>\ x^3 + a = b x</math>, with ''a'' and ''b'' positive, he would note that the maximum point of the curve <math>\ y = b x - x^3</math> occurs at <math>x = \textstyle\sqrt{\frac{b}{3}}</math>, 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.<ref>{{Citation|last=Berggren|first=J. Lennart|title=Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's ''al-Muʿādalāt|jstor=604533|journal=Journal of the American Oriental Society|volume=110|issue=2|year=1990|pages=304–309|doi=10.2307/604533|last2=Al-Tūsī|first2=Sharaf Al-Dīn|last3=Rashed|first3=Roshdi|last4=Al-Tusi|first4=Sharaf Al-Din}}</ref>
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| ===Other major figures===
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| * [['Abd al-Hamīd ibn Turk]] (fl. 830) (quadratics)
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| * [[Thabit ibn Qurra]] (826–901)
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| * [[Abū Kāmil Shujā ibn Aslam]] (c. 850 – 930) (irrationals)
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| * [[Sind ibn Ali]]
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| * [[Abū Sahl al-Qūhī]] (c. 940–1000) (centers of gravity)
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| * [[Abu'l-Hasan al-Uqlidisi]] (952 – 953) (arithmetic)
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| * [[Abu al-Saqr al-Qabisi 'Abd al-'Aziz ibn Uthman|'Abd al-'Aziz al-Qabisi]]
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| * [[Abū al-Wafā' Būzjānī]] (940 – 998) (spherical trigonometry)
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| * [[Al-Karaji]] (c. 953 – c. 1029) (algebra, induction)
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| * [[Abu Nasr Mansur]] (c. 960 – 1036) (spherical trigonometry)
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| * [[Ibn Tahir al-Baghdadi]] (c. 980–1037) (irrationals)
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| * [[Ibn al-Haytham]] (ca. 965–1040)
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| * [[Abū al-Rayḥān al-Bīrūnī]] (973 – 1048) (trigonometry)
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| * [[Omar Khayyam]] (1048–1131) (cubic equations, parallel postulate)
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| * [[Ibn Yaḥyā al-Maghribī al-Samawʾal]] (c. 1130 – c. 1180)
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| * [[Ibn Maḍāʾ]] (c. 1116 - 1196)
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| * [[Sharaf al-Dīn al-Ṭūsī]] (c. 1150–1215) (cubics)
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| * [[Naṣīr al-Dīn al-Ṭūsī]] (1201–1274) (parallel postulate)
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| * [[Jamshīd al-Kāshī]] (c. 1380–1429) (decimals and estimation of the circle constant)
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| ==See also==
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| * [[Timeline of Islamic science and technology]]
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| * [[Islamic Golden Age]]
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| * [[Hindu and Buddhist contribution to science in medieval Islam#Mathematics|Hindu and Buddhist contribution to science in medieval Islam]]
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| * [[History of geometry]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| {{refbegin|2}}
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| * {{Citation|last=Boyer|year=1991|first=Carl B.|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|chapter=Greek Trigonometry and Mensuration, and The Arabic Hegemony|edition=2nd|publisher=John Wiley & Sons|location=New York City|isbn=0-471-54397-7|ref=harv}}
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| * {{Citation|last=Katz|year=1993|first=Victor J.|authorlink=Victor J. Katz|title=A History of Mathematics: An Introduction|publisher=HarperCollins college publishers|isbn=0-673-38039-4|ref=harv}}.
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| * {{Citation|last=Ronan|year=1983|first=Colin A.|authorlink=Colin Ronan|title=The Cambridge Illustrated History of the World's Science|publisher=Cambridge University Press|isbn=0-521-25844-8|ref=harv}}
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| * {{Citation|last=Smith|year=1958|first=David E.|authorlink=David Eugene Smith|title=History of Mathematics|publisher=Dover Publications|isbn=0-486-20429-4|ref=harv}}
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| * {{Citation|last=Struik|year=1987|first=Dirk J.|authorlink=Dirk Jan Struik|title=A Concise History of Mathematics|edition=4th rev.|publisher=Dover Publications|isbn=0-486-60255-9|ref=harv}}
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| {{refend}}
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| ==Further reading==
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| {{Refbegin|2}}
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| ;Books on Islamic mathematics
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| * {{Citation|last=Berggren|first=J. Lennart|authorlink=Len Beggren|title=Episodes in the Mathematics of Medieval Islam|year=1986|publisher=Springer-Verlag|location=New York|isbn=0-387-96318-9}}
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| ** Review: {{citation|last=Toomer|first=Gerald J.|authorlink=Gerald J. Toomer|title=Episodes in the Mathematics of Medieval Islam|journal=[[American Mathematical Monthly]]|volume=95|issue=6|year=1988|doi=10.2307/2322777|page=567|publisher=Mathematical Association of America|last2=Berggren|first2=J. L.|jstor=2322777}}
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| ** Review: {{citation|first=Jan P.|last=Hogendijk|title=''Episodes in the Mathematics of Medieval Islam'' by J. Lennart Berggren|journal=Journal of the American Oriental Society|volume=109|issue=4|year=1989|pages=697–698|doi=10.2307/604119|publisher=American Oriental Society|last2=Berggren|first2=J. L.|jstor=604119}})
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| * {{Citation|last=Daffa'|first=Ali Abdullah al-|authorlink=Ali Abdullah Al-Daffa|title=The Muslim contribution to mathematics|year=1977|publisher=Croom Helm|location=London|isbn=0-85664-464-1}}
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| * {{Citation|last=Rashed|first=Roshdi|authorlink=Roshdi Rashed|others=Transl. by A. F. W. Armstrong|title=The Development of Arabic Mathematics: Between Arithmetic and Algebra|publisher=Springer|year=2001|isbn=0-7923-2565-6}}
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| * {{Citation|first=Adolf P.|last=Youschkevitch|authorlink=Adolph Pavlovich Yushkevich|coauthors=Boris A. Rozenfeld|title=Die Mathematik der Länder des Ostens im Mittelalter|year=1960|location=Berlin}} Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62–160.
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| * {{Citation|first=Adolf P.|last=Youschkevitch|title=Les mathématiques arabes: VIII<sup>e</sup>–XV<sup>e</sup> siècles|others=translated by M. Cazenave and K. Jaouiche|publisher=Vrin|location=Paris|year=1976|isbn=978-2-7116-0734-1}}
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| ; Book chapters on Islamic mathematics
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| * {{Citation |ref={{SfnRef|Berggren|2007}} |last=Berggren |first=J. Lennart |editor=Victor J. Katz |title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam | edition=Second |year=2007 |publisher=[[Princeton University Press|Princeton University]] |location=Princeton, New Jersey |isbn=978-0-691-11485-9 }}
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| * {{Citation
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| | first=Roger
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| | last=Cooke
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| | authorlink=Roger Cooke
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| | title=The History of Mathematics: A Brief Course
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| | chapter=Islamic Mathematics
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| | publisher=Wiley-Interscience
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| | year=1997
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| | isbn=0-471-18082-3
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| }}
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| ; Books on Islamic science
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| * {{Citation|first=Ali Abdullah al-|last=Daffa|first2=J.J.|last2=Stroyls|title=Studies in the exact sciences in medieval Islam|publisher=Wiley|location=New York|year=1984|isbn=0-471-90320-5}}
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| * {{Citation|first=E. S.|last=Kennedy|authorlink=Edward Stewart Kennedy|title=Studies in the Islamic Exact Sciences|year=1984|publisher=Syracuse Univ Press|isbn=0-8156-6067-7}}
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| ; Books on the history of mathematics
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| * {{Citation|last=Joseph|first=George Gheverghese|authorlink=George Gheverghese Joseph|title=The Crest of the Peacock: Non-European Roots of Mathematics|edition=2nd|publisher=Princeton University Press|year=2000|isbn=0-691-00659-8}} (Reviewed: {{citation|first=Victor J.|last=Katz|title=''The Crest of the Peacock: Non-European Roots of Mathematics'' by George Gheverghese Joseph|journal=The College Mathematics Journal|volume=23|issue=1|year=1992|pages=82–84|doi=10.2307/2686206|publisher=Mathematical Association of America|last2=Joseph|first2=George Gheverghese|jstor=2686206}})
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| * {{Citation|last=Youschkevitch|first=Adolf P.|title=Gesichte der Mathematik im Mittelalter|publisher=BG Teubner Verlagsgesellschaft|location=Leipzig|year=1964}}
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| ;Journal articles on Islamic mathematics
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| * [[Jens Høyrup|Høyrup, Jens]]. [http://akira.ruc.dk/~jensh/Publications/1987_Formation%20of%20Islamic%20mathematics.PDF “The Formation of «Islamic Mathematics»: Sources and Conditions”]. ''Filosofi og Videnskabsteori på Roskilde Universitetscenter''. 3. Række: ''Preprints og Reprints'' 1987 Nr. 1.
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| ;Bibliographies and biographies
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| * [[Carl Brockelmann|Brockelmann, Carl]]. ''Geschichte der Arabischen Litteratur''. 1.–2. Band, 1.–3. Supplementband. Berlin: Emil Fischer, 1898, 1902; Leiden: Brill, 1937, 1938, 1942.
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| * {{Citation|last=Sánchez Pérez|first=José A.|authorlink=José Augusto Sánchez Pérez|title=Biografías de Matemáticos Árabes que florecieron en España|location=Madrid|publisher=Estanislao Maestre|year=1921}}
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| * {{Citation|last=Sezgin|first=Fuat|authorlink=Fuat Sezgin|title=Geschichte Des Arabischen Schrifttums|publisher=Brill Academic Publishers|language=German|year=1997|isbn=90-04-02007-1}}
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| * {{Citation|last=Suter|first=Heinrich|authorlink=Heinrich Suter|title=Die Mathematiker und Astronomen der Araber und ihre Werke|series=Abhandlungen zur Geschichte der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen, X Heft|location=Leipzig|year=1900}}
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| ; Television documentaries
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| * [[Marcus du Sautoy]] (presenter) (2008). "The Genius of the East". ''[[The Story of Maths]]''. [[BBC]].
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| * [[Jim Al-Khalili]] (presenter) (2010). ''[[Science and Islam (documentary)|Science and Islam]]''. [[BBC]].
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| {{Refend}}
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| == External links ==
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| * {{cite web|last=Hogendijk|first=Jan P.|date=January 1999|year=1999|url=http://www.jphogendijk.nl/publ/Islamath.html|title=Bibliography of Mathematics in Medieval Islamic Civilization|ref=harv}}
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| * {{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}}
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| * [http://www.saudiaramcoworld.com/issue/200703/rediscovering.arabic.science.htm Richard Covington, ''Rediscovering Arabic Science'', 2007, Saudi Aramco World]
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| {{Islamic mathematics}}
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| {{DEFAULTSORT:Mathematics In Medieval Islam}}
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| [[Category:Islamic mathematics| ]]
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| [[Category:Islamic Golden Age]]
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