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 |
Abū Bakr Muḥammad ibn al Ḥasan al-Karajī (Persian:ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-centuryPersian[2][3][4]mathematician andengineer who flourished atBaghdad. He was born inKaraj,[1] a city nearTehran. 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), andAl-Kafi fi'l-hisab (Sufficient on calculation).
Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced byDiophantus) 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 bookal-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 formonomials 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 onalgebra andpolynomials 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 thebinomial coefficients and the first description ofPascal'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 byal-Samaw'al, Al-Karaji introduced the idea of argument bymathematical 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 toAryabhata [...] Al-Karaji did not, however, state a general result for arbitraryn. 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 thetruth of the statement forn = 1 (1 = 13) and the deriving of the truth forn =k from that ofn =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 fromn = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument inal-Fakhri is the earliest extant proof ofthe sum formula for integral cubes.[18]
Of Persian origin, he spent an important part of his scientific life in Baghdad where he composed ground breaking mathematical books.
Al-Karajī Abū Bakr Muh.ammad was a Persian mathematician and engineer.
During the tenth century CE, the Iranian mathematician al-Karaji (...)
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.
He was also the first to use the method of proof by mathematical induction to prove his results, which he also used to prove the sum formula for integral cubes, an important result in integral calculus.
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