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Persian mathematician and engineer (c. 953 – c. 1029) From Wikipedia, the free encyclopedia
Abū Bakr Muḥammad ibn al Ḥasan al-Karajī (Persian: ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian[2][3][4] mathematician and engineer who flourished at Baghdad. He was born in Karaj,[1] a city near 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).
Abū Bakr al-Karajī | |
|---|---|
Diagrams from Al-Karaji's work on "hidden waters" | |
| Born | 953 |
| Died | 1029 (aged 75–76) |
| Nationality | Persian |
Main interests | Mathematics, Engineering |
Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by Diophantus) but most regard him as more original,[5] 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.[6]
He expounded the basic principles of hydrology[7] and this book reveals his profound knowledge of this science and has been described as the oldest extant text in this field.[8][9][10]
He systematically studied the algebra of exponents, and was the first to define the rules for monomials like x,x²,x³ and their reciprocals in the cases of multiplication and division. 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.[11]
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.
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.[6][12]
Al-Karaji gave an early formulation of the binomial coefficients and the first description of Pascal's triangle.[13][14][15] He is also presumed to have discovered the binomial theorem.[16]
In a now lost work known only from subsequent quotation by al-Samaw'al, Al-Karaji introduced the idea of argument by mathematical induction.[17] 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 = 13) 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.[18]
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