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Abū Bakr al-Karajī
Muhammad al karaji 01.jpg
Diagrams from Al-Karaji's work on "hidden waters"
Born 953
Karaj, Persia
Died 1029 (aged 75–76)
Nationality Persian
Main interests
Mathematics, Engineering

Abū Bakr ibn Muḥammad ibn al Ḥusayn al-Karajī (Persian: ابوبکر محمد بن حسین کرجی‎; c. 953 – c. 1029) was a 10th-century Persian[1][2][3] mathematician and engineer who flourished at Baghdad. He was born in Karaj, 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).


Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by Diophantus)[4] 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.

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 Galileo Galilei, Johannes Kepler or Isaac Newton, but long after Aristotle and Ptolemy. This book reveals a profound knowledge of hydrology and has been described as the oldest text of its kind in this field.[5]

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.[6]

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.[7][8]

Al-Karaji gave the first formulation of the binomial theorem[9] and the first description of Pascal's triangle.[10][7]

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 = 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.[11]

See also[edit]


  1. ^ "Muhammad Al-Karaji: A Mathematician Engineer from the Early 11th Century | Muslim Heritage". Retrieved 2018-08-10. Of Persian origin, he spent an important part of his scientific life in Baghdad where he composed ground breaking mathematical books. 
  2. ^ Selin, Helaine (2008). Encyclopaedia of the history of science, technology, and medicine in non-western cultures. Berlin New York: Springer. p. 131. ISBN 9781402049606. Al-Karajī Abū Bakr Muh.ammad was a Persian mathematician and engineer. 
  3. ^ Meri, Josef W. (January 2006). Medieval Islamic Civilization, Volume 1 An Encyclopedia. Routledge. p. 32. ISBN 978-0-415-96691-7. During the tenth century CE, the Iranian mathematician al-Karaji (...) 
  4. ^
  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."
  6. ^ Katz, History of Mathematics, first edition, p237
  7. ^ a b O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji", MacTutor History of Mathematics archive, University of St Andrews .
  8. ^ "You Have Got to Know...Mathematics" "Page 26"
  9. ^ The developpement of Arabic Mathematics Between Arithmetic and Algebra - R. Rashed "Page 63"
  10. ^ Encyclopedia Britannica "Al-Karajī, also known as al-Karkhī, in full, Abū Bakr ibn Muḥammad ibn al-Ḥusayn al-Karajī, (born c. 980, most likely Karaj, Persia, rather than Karkh, near Baghdad, Iraq—died c. 1030), mathematician and engineer who held an official position in Baghdad (c. 1010–1015), perhaps culminating in the position of vizier, during which time he wrote his three main works, al-Fakhrī fīʾl-jabr wa’l-muqābala (“Glorious on algebra”), al-Badī‘ fī’l-hisāb (“Wonderful on calculation”), and al-Kāfī fī’l-hisāb (“Sufficient on calculation”). A now lost work of his contained the first description of what later became known as Pascal’s triangle."
  11. ^ Katz (1998), p. 255

References and external links[edit]