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Abū Bakr ibn Muḥammad ibn al Ḥusayn al-Karajī (or al-Karkhī) (c. 953 – c. 1029) was a 10th-century mathematician and engineer who flourished at Baghdad. 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).
There is ambiguity in what his last name was. Some medieval Arabic documents have al-Karajī and others have al-Karkhī.^{[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 al-Karkhī, indicating that he was born in Karkh, a suburb of Baghdad, or al-Karajī 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.
Al-Karaji 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 al-fakhri fi al-jabr wa al-muqabala, 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 square-cube, 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 al-Samaw'al Al-Karaji introduced the idea of argument by mathematical induction. As Katz says
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 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, 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 the sum formula for integral cubes.^{[4]}
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