View source for Biography:Al-Karaji

{{Infobox philosopher
|name        = Abū Bakr al-Karajī
|caption     = Diagrams from Al-Karaji's work on "hidden waters"
|image       = Muhammad al karaji 01.jpg
|birth_date  = 953
|birth_place = [[Place:Karaj|Karaj]], Persia
|death_place = 
|nationality = Persian
|school_tradition = 
|main_interests = [[Mathematics]], [[Engineering:Engineering|Engineering]]
|influences = [[Biography:Diophantus|Diophantus]], [[Biography:Abū Kāmil Shujāʿ ibn Aslam|Abū Kāmil Shujāʿ ibn Aslam]]
|influenced = 
|pronunciation = 
}}

'''{{transl|ar|ALA|Abū Bakr Muḥammad ibn al Ḥasan al-Karajī}}''' ({{lang-fa|ابو بکر محمد بن الحسن الکرجی}}; c. 953 &ndash; c. 1029) was a 10th-century Persian<ref>{{Cite web|url=http://www.muslimheritage.com/article/muhammad-al-karaji|title=Muhammad Al-Karaji: A Mathematician Engineer from the Early 11th Century {{!}} Muslim Heritage|website=www.muslimheritage.com|language=en|access-date=2018-08-10|quote=Of Persian origin, he spent an important part of his scientific life in Baghdad where he composed ground breaking mathematical books.}}</ref><ref>{{cite book | last = Selin | first = Helaine | title = Encyclopaedia of the history of science, technology, and medicine in non-western cultures | publisher = Springer | location = Berlin New York | year = 2008 | isbn = 9781402049606 | page=131 | quote=Al-Karajī Abū Bakr Muh.ammad was a Persian mathematician and engineer.}}</ref><ref>{{cite book|last1=Meri|first1=Josef W.|title=Medieval Islamic Civilization, Volume 1  An Encyclopedia|date=January 2006|publisher=Routledge|isbn=978-0-415-96691-7|page=32|quote=During the tenth century CE, the Iranian mathematician al-Karaji (...)}}</ref> [[Biography:Mathematician|mathematician]] and [[Engineering:Engineer|engineer]] who flourished at [[Place:Baghdad|Baghdad]]. He was born in [[Place:Karaj|Karaj]], a city near [[Place:Tehran|Tehran]]. His three principal surviving works are mathematical: ''Al-Badi' fi'l-hisab'' (''Wonderful on calculation''), ''Al-Fakhri fi'l-jabr wa'l-muqabala'' (''Glorious on algebra''), and ''Al-Kafi fi'l-hisab'' (''Sufficient on calculation'').

== Work ==

Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by [[Biography:Diophantus|Diophantus]]) but most regard him as more original,<ref>http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Karaji.html</ref> in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book ''al-fakhri fi al-jabr wa al-muqabala'', which survives from the medieval era in at least four copies.<ref name=Karaji>{{MacTutor|id=Al-Karaji|title=Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji}}</ref>

In his book "Extraction of hidden waters" he has mentioned that earth is spherical in shape but considers it the centre of the universe long before [[Biography:Galileo Galilei|Galileo Galilei]], [[Biography:Johannes Kepler|Johannes Kepler]] or [[Biography:Isaac Newton|Isaac Newton]], but long after [[Biography:Aristotle|Aristotle]] and [[Biography:Ptolemy|Ptolemy]]. He expounded the basic principles of hydrology<ref>{{Cite book|url=https://books.google.fr/books?id=DBRpDgAAQBAJ&pg=PA19&lpg=PA19&dq=al+karaji+hydrology&source=bl&ots=kO12dVE_ht&sig=ACfU3U0uFuuk2xK-9NeOnVX2tu2bGMK64Q&hl=fr&sa=X&ved=2ahUKEwjRse3nz6rgAhXOxYUKHeGXD1wQ6AEwDXoECAgQAQ#v=onepage&q=al%20karaji%20hydrology&f=false|title=Hydrology: Principles and Processes|last=Robinson|first=M.|last2=Ward|first2=R. C.|date=2017-02-15|publisher=IWA Publishing|isbn=9781780407289|language=en|page=19}}</ref> and this book reveals a profound knowledge of this science and has been described as the oldest extant text in this field.<ref>[http://www.muslimheritage.com/article/muhammad-al-karaji#sec_5/ Muslim Heritage, Mohammed Abattouy] " Al-Karaji is also the author of Inbat al-miyah al-khafiya (The Extraction of Hidden Waters), a technical treatise that reveals such a profound knowledge of hydrology that it should be celebrated as the oldest text of its kind in this field."</ref><ref>{{Cite book|url=https://books.google.fr/books?id=IQlQDwAAQBAJ&pg=PA37&lpg=PA37&dq=al+karaji+hydrology&source=bl&ots=N6xj7Mcz95&sig=ACfU3U124aW9gv3iynpIdW-p21eEP_sAUw&hl=fr&sa=X&ved=2ahUKEwjm5LCf0qrgAhWH1-AKHWwqDYE4ChDoATACegQIBBAB#v=onepage&q=al%20karaji%20hydrology&f=false|title=Tectonic Evolution, Collision, and Seismicity of Southwest Asia: In Honor of Manuel Berberian's Forty-Five Years of Research Contributions|last=Sorkhabi|first=Rasoul|date=2017-12-21|publisher=Geological Society of America|isbn=9780813725253|language=en|page=37}}</ref><ref>{{Cite journal|last=Niazi|first=Kaveh|date=2016-01-01|title=Karajī’s Discourse on Hydrology|url=https://brill.com/abstract/journals/orie/44/1-2/article-p44_2.xml|journal=Oriens|language=en|volume=44|issue=1-2|pages=44–68|doi=10.1163/18778372-04401003|issn=0078-6527|quote=The hydrological concepts presented in Inbāṭ al-miyāh al-khafīya, Muḥammad Karajī’s 11th century text on the construction of the qanāt, contain unexpected premises and theories that set this text apart from its contemporaries. Even when not straying far from the Aristotelian cosmology of the medieval world, Karajī’s hydrological discussions often represent a fresh take on the common scientific wisdom regarding the flow of water at and near the earth’s surface. }}</ref>

He systematically studied the algebra of exponents, and was the first to realise that the sequence x, x^2, x^3,... could be extended indefinitely; and the reciprocals 1/x, 1/x^2, 1/x^3,... . However, since for example the product of a square and a cube would be expressed, in words rather than in numbers, as a square-cube, the numerical property of adding exponents was not clear.<ref>Katz, History of Mathematics, first edition, p237</ref>

His work on [[Algebra|algebra]] and [[Polynomial|polynomial]]s gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials; though he was restricted to dividing polynomials by monomials.

F. Woepcke was the first historian to realise the importance of al-Karaji's work and later historians mostly agree with his interpretation. He praised Al-Karaji for being the first who introduced the theory of algebraic calculus.<ref name=Karaji/><ref>[https://books.google.fr/books?id=CdIAAwAAQBAJ&pg=PA26&lpg=PA26&dq=al+karaji+first+formulation+of+the+binomial+theorem&source=bl&ots=pQINDvdPxZ&sig=ZqTfwF_lrERYHijT5uz-c_mcn-0&hl=fr&sa=X&ved=0ahUKEwi7v4n3lO3aAhXKvRQKHfP2COc4ChDoAQg_MAI#v=onepage&q=al%20karaji%20first%20formulation%20of%20the%20binomial%20theorem&f=false/ "You Have Got to Know...Mathematics"] "Page 26"</ref>

Al-Karaji gave the first formulation of the [[Binomial coefficient|binomial coefficient]]s and the first description of [[Pascal's triangle]].<ref>{{Cite book|url=https://books.google.fr/books?id=kAjABAAAQBAJ&pg=PA54&lpg=PA54&dq=al+karaji+binomial+theorem&source=bl&ots=Al4S22US_g&sig=35YJq4ATbr86k1XTgdzgTvCVsbc&hl=fr&sa=X&redir_esc=y#v=onepage&q=al%20karaji%20binomial%20theorem&f=false|title=From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren|last=Sidoli|first=Nathan|last2=Brummelen|first2=Glen Van|date=2013-10-30|publisher=Springer Science & Business Media|isbn=9783642367366|language=en|page=54}}</ref><ref>{{Cite book|url=https://books.google.fr/books?id=kt9DIY1g9HYC&pg=PA132&lpg=PA132&dq=al+karaji+pascal's+triangle&source=bl&ots=NN84FLKEdM&sig=VaHrwEYBqKKo1DC5Vb9xdarXocw&hl=fr&sa=X&ved=2ahUKEwiis4CR497fAhVHSxoKHciHCzI4ChDoATAJegQIABAB#v=onepage&q=al%20karaji%20pascal's%20triangle&f=false|title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures|last=Selin|first=Helaine|date=2008-03-12|publisher=Springer Science & Business Media|isbn=9781402045592|language=en|page=132}}</ref><ref>[https://books.google.fr/books?id=vSkClSvU_9AC&pg=PA62&redir_esc=y#v=onepage&q&f=false/ The Development of Arabic Mathematics Between Arithmetic and Algebra - R. Rashed] "Page 63"</ref> He is also credited with the discovery of the binomial theorem.<ref>{{Cite web|url=https://core.ac.uk/download/pdf/82000184.pdf|website=core.ac.uk|access-date=2019-01-08|title=THE BINOMIAL THEOREM : A WIDESPREAD CONCEPT IN MEDIEVAL ISLAMIC MATHEMATICS|page=401}}</ref>
In a now lost work known only from subsequent quotation by al-Samaw'al Al-Karaji introduced the idea of argument by [[Mathematical induction|mathematical induction]]. As Katz says {{quote|text=Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitrary ''n''. He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer. [...] Al-Karaji's argument includes in essence the two basic components of a modern argument by induction, namely the [[Truth|truth]] of the statement for ''n'' = 1 (1 = 1<sup>3</sup>) and the deriving of the truth for ''n'' = ''k'' from that of ''n'' = ''k'' - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from ''n'' = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in ''al-Fakhri'' is the earliest extant proof of [[Squared triangular number|the sum formula for integral cubes]].<ref>Katz (1998), p. 255</ref>}}

==See also==
*Mathematics in medieval Islam
*Science in medieval Islam
*List of Iranian scientists

== Notes==
{{reflist}}

== References and external links==
* {{MacTutor|id=Al-Karaji|title=Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji}}
* {{DSB|first=Roshdi|last=Rashed|url=http://www.encyclopedia.com/doc/1G2-2830902256.html|title=Al-Karajī (or Al-Karkhī), Abū Bakr Ibn Muḥammad Ibn al Ḥusayn}}
* J. Christianidis. ''Classics in the History of Greek Mathematics'', p.&nbsp;260
* Carl R. Seaquist, Padmanabhan Seshaiyer, and Dianne Crowley. [https://web.archive.org/web/20060909142841/http://www.cs.southwestern.edu/txcmj/aculture4f.pdf "Calculation across Cultures and History"] (''Texas College Mathematics Journal'' 1:1, 2005; pp 15&ndash;31) [PDF]
* Matthew Hubbard and Tom Roby. [https://web.archive.org/web/20060912173217/http://binomial.csuhayward.edu/MidEast.html "The History of the Binomial Coefficients in the Middle East"](from "Pascal's Triangle from Top to Bottom") 
* Fuat Sezgin. ''Geschichte des arabischen Schrifttums'' (1974, Leiden: E. J. Brill)
* James J. Tattersall. ''Elementary Number Theory in Nine Chapters'', p.&nbsp;32
* [[Biography:Mariusz Wodzicki|Mariusz Wodzicki]]. [http://math.berkeley.edu/~wodzicki/160.F05/HistIntr.pdf "Early History of Algebra: a Sketch"] (''Math'' 160, Fall 2005) [PDF]
* [https://www.webcitation.org/5PuedD0UI?url=http://0-www.search.eb.com.library.uor.edu/eb/article-9343818 "al-Karaji"] &mdash; ''Encyclopædia Britannica'' Online (4 April 2006)
* [https://archive.org/details/extraitdufakhrt00karagoog Extrait du ''Fakhri'', traité d'Algèbre par Abou Bekr Mohammed Ben Alhaçan Alkarkhi], presented with commentary by F. Woepcke, year 1853.

{{Islamic mathematics}}


{{DEFAULTSORT:Karaji}}
[[Category:10th-century mathematicians]]
[[Category:11th-century mathematicians]]

{{Sourceattribution|Al-Karaji|1}}