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AlKaraji
Abū Bakr ibn Muḥammad ibn al Ḥusayn alKarajī (or alKarkhī) (c. 953 – c. 1029) was a 10thcentury mathematician and engineer who flourished at Baghdad. His three principal surviving works are mathematical: AlBadi' fi'lhisab (Wonderful on calculation), AlFakhri fi'ljabr wa'lmuqabala (Glorious on algebra), and AlKafi fi'lhisab (Sufficient on calculation).
Name
There is ambiguity in what his last name was. Some medieval Arabic documents have alKarajī and others have alKarkhī.^{[1]} Arabic documents from the Baghdad of that era are sometimes written without diacritical points, whereby the written name is inherently ambiguous and can be read in Arabic as Karajī (reading ج) or Karkhī (reading خ) or Karahī or Karhī (reading ح)  see Arabic rasm notation, i.e. the absence of i'jam diacritic distinctions of consonants. His name could have been alKarkhī, indicating that he was born in Karkh, a suburb of Baghdad, or alKarajī indicating his family came from the city of Karaj in Iran. He certainly lived and worked for most of his life in Baghdad, however, which was the scientific and trade capital of the Islamic world.
Work
AlKaraji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by Diophantus)^{[2]} but most regard him as more original, in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book alfakhri fi aljabr wa almuqabala, which survives from the medieval era in at least four copies.^{[1]}
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 squarecube, the numerical property of adding exponents was not clear.^{[3]}
His work on algebra and polynomials gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials; though he was restricted to dividing polynomials by monomials.
He wrote on the binomial theorem and Pascal's triangle.
In a now lost work known only from subsequent quotation by alSamaw'al AlKaraji introduced the idea of argument by mathematical induction. As Katz saysAnother important idea introduced by alKaraji and continued by alSamaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus alKaraji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] AlKaraji 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. [...] AlKaraji's argument includes in essence the two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 1^{3}) 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, alKaraji'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 alFakhri is the earliest extant proof of the sum formula for integral cubes.^{[4]}
See also
Notes
 ^ ^{a} ^{b} Template:DSB
 ^ http://wwwhistory.mcs.stand.ac.uk/history/Biographies/AlKaraji.html
 ^ Kats, History of Mathematics, first edition, p237
 ^ Katz (1998), p. 255
References and external links
 O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn alHusayn AlKaraji", MacTutor History of Mathematics archive, University of St Andrews.
 Template:DSB
 J. Christianidis. Classics in the History of Greek Mathematics, p. 260
 Carl R. Seaquist, Padmanabhan Seshaiyer, and Dianne Crowley. "Calculation across Cultures and History" (Texas College Mathematics Journal 1:1, 2005; pp 15–31) [PDF]
 Matthew Hubbard and Tom Roby. "The History of the Binomial Coefficients in the Middle East" (from "Pascal's Triangle from Top to Bottom")^{[dead link]}
 Fuat Sezgin. Geschichte des arabischen Schrifttums (1974, Leiden: E. J. Brill)
 James J. Tattersall. Elementary Number Theory in Nine Chapters, p. 32
 Mariusz Wodzicki. "Early History of Algebra: a Sketch" (Math 160, Fall 2005) [PDF]
 "alKaraji" — Encyclopædia Britannica Online (4 April 2006)
 Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhaçan Alkarkhi, presented with commentary by F. Woepcke, year 1853.
