Editing History of mathematics (section)

== Islamic empires ==
{{Main|Mathematics in medieval Islam}}
{{See also|History of the Hindu–Arabic numeral system}}
[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|Page from ''[[The Compendious Book on Calculation by Completion and Balancing]]'' by [[Muhammad ibn Mūsā al-Khwārizmī]] (c.&nbsp;AD 820)]]
The [[Caliphate|Islamic Empire]] established across the [[Middle East]], [[Central Asia]], [[North Africa]], [[Iberian Peninsula|Iberia]], and in parts of [[History of India|India]] in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in [[Arabic language|Arabic]], they were not all written by [[Arab]]s, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time.<ref>Abdel Haleem, Muhammad A. S. "The Semitic Languages", https://doi.org/10.1515/9783110251586.811, "Arabic became the language of scholarship in science and philosophy in the 9th century when the ‘translation movement’ saw concerted work on translations of Greek, Indian, Persian and Chinese, medical, philosophical and scientific texts", p. 811.</ref>

In the 9th century, the Persian mathematician [[Muḥammad ibn Mūsā al-Khwārizmī]] wrote an important book on the [[Hindu–Arabic numerals]] and one on methods for solving equations. His book ''On the Calculation with Hindu Numerals'', written about 825, along with the work of [[Al-Kindi]], were instrumental in spreading [[Indian mathematics]] and [[Hindu–Arabic numeral system|Indian numerals]] to the West. The word ''[[algorithm]]'' is derived from the Latinization of his name, Algoritmi, and the word ''algebra'' from the title of one of his works, ''[[The Compendious Book on Calculation by Completion and Balancing|Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala]]'' (''The Compendious Book on Calculation by Completion and Balancing''). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,<ref>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 230}} "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwārizmī's exposition that his readers must have had little difficulty in mastering the solutions."</ref> and he was the first to teach algebra in an [[Elementary algebra|elementary form]] and for its own sake.<ref>Gandz and Saloman (1936). "The sources of Khwarizmi's algebra", ''Osiris'' i, pp. 263–77: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".</ref> He also discussed the fundamental method of "[[Reduction (mathematics)|reduction]]" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as ''al-jabr''.<ref name=Boyer-229>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 229}} "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."</ref> His algebra was also no longer concerned "with a series of problems to be resolved, but an [[Expository writing|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."<ref name=Rashed-Armstrong>{{Cite book | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=978-0-7923-2565-9 | oclc=29181926 | pages=11–12}}</ref>

In Egypt, [[Abu Kamil]] extended algebra to the set of [[irrational numbers]], accepting square roots and fourth roots as solutions and coefficients to quadratic equations. He also developed techniques used to solve three non-linear simultaneous equations with three unknown variables. One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found 2676 solutions.<ref name="HSTM">{{Cite encyclopedia | publisher = Springer|  pages = 4–5| last = Sesiano| first = Jacques| title = Abū Kāmil | encyclopedia = Encyclopaedia of the history of science, technology, and medicine in non-western cultures| year= 1997}}</ref> His works formed an important foundation for the development of algebra and influenced later mathematicians, such as al-Karaji and Fibonacci.

Further developments in algebra were made by [[Al-Karaji]] in his treatise ''al-Fakhri'', where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a [[Mathematical proof|proof]] by [[mathematical induction]] appears in a book written by Al-Karaji around 1000 AD, who used it to prove the [[binomial theorem]], [[Pascal's triangle]], and the sum of integral [[Cube (algebra)|cubes]].<ref>{{Harv|Katz|1998|loc=pp. 255–59}}</ref> The [[historian]] of mathematics, F. Woepcke,<ref>Woepcke, F. (1853). ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi''. [[Paris]].</ref> praised Al-Karaji for being "the first who introduced the [[theory]] of [[algebra]]ic [[calculus]]." Also in the 10th century, [[Abul Wafa]] translated the works of [[Diophantus]] into Arabic. [[Ibn al-Haytham]] was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a [[paraboloid]], and was able to generalize his result for the integrals of [[polynomial]]s up to the [[Quartic polynomial|fourth degree]]. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.<ref name=Katz>{{cite journal | last1 = Katz | first1 = Victor J. | year = 1995 | title = Ideas of Calculus in Islam and India | journal = Mathematics Magazine | volume = 68 | issue = 3| pages = 163–74 | doi=10.2307/2691411| jstor = 2691411 }}</ref>

In the late 11th century, [[Omar Khayyam]] wrote ''Discussions of the Difficulties in Euclid'', a book about what he perceived as flaws in [[Euclid's Elements|Euclid's ''Elements'']], especially the [[parallel postulate]]. He was also the first to find the general geometric solution to [[cubic equation]]s. He was also very influential in [[calendar reform]].<ref>{{Cite journal|last=Alam|first=S|year=2015|title=Mathematics for All and Forever|url=http://www.iisrr.in/mainsite/wp-content/uploads/2015/01/IISRR-IJR-1-Mathematics-for-All-...-Syed-Samsul-Alam.pdf|journal=Indian Institute of Social Reform & Research International Journal of Research}}</ref>

In the 13th century, [[Nasir al-Din Tusi]] (Nasireddin) made advances in [[spherical trigonometry]]. He also wrote influential work on Euclid's [[parallel postulate]]. In the 15th century, [[Ghiyath al-Kashi]] computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating ''n''th roots, which was a special case of the methods given many centuries later by [[Paolo Ruffini (mathematician)|Ruffini]] and [[William George Horner|Horner]].

Other achievements of Muslim mathematicians during this period include the addition of the [[decimal point]] notation to the [[Arabic numerals]], the discovery of all the modern [[trigonometric function]]s besides the sine, [[al-Kindi]]'s introduction of [[cryptanalysis]] and [[frequency analysis]], the development of [[analytic geometry]] by [[Ibn al-Haytham]], the beginning of [[algebraic geometry]] by [[Omar Khayyam]] and the development of an [[Mathematical notation|algebraic notation]] by [[Abū al-Hasan ibn Alī al-Qalasādī|al-Qalasādī]].<ref name=Qalasadi>{{MacTutor Biography|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref>

During the time of the [[Ottoman Empire]] and [[Safavid Empire]] from the 15th century, the development of Islamic mathematics became stagnant.