View source for 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 of Perga|Apollonius]]) and [[Indian mathematics]] ([[Aryabhata]], [[Brahmagupta]]), saw important developments, such as the full development of the decimal [[place-value system]] to include [[decimal fractions]], the first systematised study of [[algebra]] (named for [[The Compendious Book on Calculation by Completion and Balancing|the work]] of  scholar [[Al-Kwarizmi]]), and advances in [[geometry]] and [[trigonometry]].

Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries.

(TO DO: fix math, expand, organize, cross-reference, illustrate.)

== History ==

=== Algebra ===

The study of [[algebra]], which itself is an [[Arabic language|Arabic]] word meaning "reunion of broken parts", flourished during the Islamic golden age. Al-Khwarizmi is, along with the [[Greek people|Greek]] mathematician [[Diophantus]], known as the father of algebra.

In his book ''[[The Compendious Book on Calculation by Completion and Balancing]]'' Al-Khwarizmi deals with ways to solve for the [[positive number#Terminology for signs|positive]] [[Properties of polynomial roots|root]]s of first and second degree (linear and quadratic) [[polynomial equations]].

He also introduces the method of [[Reduction (mathematics)|reduction]], and unlike Diophantus, gives general solutions for the equations he deals with.<ref name="fabpedigree.com"/>

Al-Khwarizmi'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 al-Banna' al-Marrakushi]] and [[Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī]].

Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. [[Omar Khayyam]], along with [[Sharaf al-Dīn al-Tūsī]], 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 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.

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]].

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.

He revised [[Ptolemy]]'s ''[[Geography (Ptolemy)|Geography]]'' and wrote on astronomy and astrology.

=== Induction ===

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).

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]].

== Major figures and developments ==

=== Omar Khayyám ===

[[Omar Khayyám]] (c. 1038/48 in [[Iran]] – 1123/24) 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]].

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, but they did not generalize the method to cover all equations with positive [[Zero of a function|roots]].

=== Sharaf al-Dīn al-Ṭūsī ===

[[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>

=== Other major figures ===

* [['Abd al-Hamīd ibn Turk]] (fl. 830) (quadratics)
* [[Thabit ibn Qurra]] (826–901)
* [[Abū Kāmil Shujā ibn Aslam]] (c. 850 – 930) (irrationals)
* [[Sind ibn Ali]]
* [[Abū Sahl al-Qūhī]] (c. 940–1000) (centers of gravity)
* [[Abu'l-Hasan al-Uqlidisi]] (952 – 953) (arithmetic)
* [[Abu al-Saqr al-Qabisi 'Abd al-'Aziz ibn Uthman|'Abd al-'Aziz al-Qabisi]]
* [[Abū al-Wafā' Būzjānī]] (940 – 998) (spherical trigonometry)
* [[Al-Karaji]] (c. 953 – c. 1029) (algebra, induction)
* [[Abu Nasr Mansur]] (c. 960 – 1036) (spherical trigonometry)
* [[Ibn Tahir al-Baghdadi]] (c. 980–1037) (irrationals)
* [[Ibn al-Haytham]] (ca. 965–1040)
* [[Abū al-Rayḥān al-Bīrūnī]] (973 – 1048) (trigonometry)
* [[Ibn Yaḥyā al-Maghribī al-Samawʾal]] (c. 1130 – c. 1180)
* [[Ibn Maḍāʾ]] (c. 1116 - 1196)
* [[Naṣīr al-Dīn al-Ṭūsī]] (1201–1274) (parallel postulate)
* [[Jamshīd al-Kāshī]] (c. 1380–1429) (decimals and estimation of the circle constant)

== See also ==

* [[Arabic numerals]]
* [[History of algebra]]
* [[History of calculus]]
* [[History of geometry]]
* [[History of trigonometry]]
* [[Indian influence on Islamic science#Mathematics|Indian influence on Islamic mathematics in medieval Islam]]
* [[Islamic Golden Age]]
* [[Mathematics]]
* [[Timeline of Islamic science and technology]]

== External links ==

* [https://en.wikipedia.org/wiki/Mathematics_in_medieval_Islam Mathematics in medieval Islam]
* {{cite web|last=Hogendijk|first=Jan P.|date=January 1999|url=http://www.jphogendijk.nl/publ/Islamath.html|title=Bibliography of Mathematics in Medieval Islamic Civilization|ref=harv}}
* {{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}}
* [http://www.saudiaramcoworld.com/issue/200703/rediscovering.arabic.science.htm Richard Covington, ''Rediscovering Arabic Science'', 2007, Saudi Aramco World]