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Abū Bakr ibn Muḥammad ibn al Ḥusayn al-Karajī (or al-Karkhī) (c. 953 in Karaj or Karkh – c. 1029) was a 10th-century Persian 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).

Name

There is ambiguity in what his last name was. Some medieval Arabic documents have al-Karajī and others have al-Karkhī. 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 Karahi or Karhi (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. 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

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, 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.

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.

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) 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.

See also

• Mathematics in medieval Islam
• Science in medieval Islam
• List of Iranian scientists

Notes

1. Studies on Greek Algebra, Jacques Sesiano, Classics In The History Of Greek Mathematics, ed. Jean Christianidis, (Kluwer Academic Publishers, 2004), 260.
2. ^ Rashed, Roshdi (1970–1980). "Al-Karajī (or Al-Karkhī), Abū Bakr Ibn Muḥammad Ibn al Ḥusayn". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. ISBN 978-0-684-10114-9.
3. http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Karaji.html
4. Kats, History of Mathematics, first edition, p237
5. Katz (1998), p. 255

References and external links

• 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
• Rashed, Roshdi (1970–1980). "Al-Karajī (or Al-Karkhī), Abū Bakr Ibn Muḥammad Ibn al Ḥusayn". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. ISBN 978-0-684-10114-9.
• 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)
• Matthew Hubbard and Tom Roby. "The History of the Binomial Coefficients in the Middle East" (from "Pascal's Triangle from Top to Bottom")
• 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)
• "al-Karaji" — 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.

v t e Mathematics in the medieval Islamic world
Mathematicians	9th century 'Abd al-Hamīd ibn Turk Sanad ibn Ali al-Jawharī Al-Ḥajjāj ibn Yūsuf Al-Kindi Qusta ibn Luqa Al-Mahani al-Dinawari Banū Mūsā brothers Hunayn ibn Ishaq Al-Khwarizmi Yusuf al-Khuri Ishaq ibn Hunayn Na'im ibn Musa Thābit ibn Qurra al-Marwazi Abu Said Gorgani 10th century Abu al-Wafa al-Khazin Al-Qabisi Abu Kamil Ahmad ibn Yusuf Aṣ-Ṣaidanānī Sinān ibn al-Fatḥ al-Khojandi Al-Nayrizi Al-Saghani Brethren of Purity Ibn Sahl Ibn Yunus al-Uqlidisi Al-Battani Sinan ibn Thabit Ibrahim ibn Sinan Al-Isfahani Nazif ibn Yumn al-Qūhī Abu al-Jud Al-Sijzi Al-Karaji al-Majriti al-Jabali 11th century Abu Nasr Mansur Alhazen Kushyar Gilani Al-Biruni Ibn al-Samh Abu Mansur al-Baghdadi Avicenna al-Jayyānī al-Nasawī al-Zarqālī ibn Hud Al-Isfizari Omar Khayyam Muhammad al-Baghdadi 12th century Jabir ibn Aflah Al-Kharaqī Al-Khazini Al-Samawal al-Maghribi al-Hassar Sharaf al-Din al-Tusi Ibn al-Yasamin 13th century Ibn al‐Ha'im al‐Ishbili Ahmad al-Buni Ibn Munim Alam al-Din al-Hanafi Ibn Adlan al-Urdi Nasir al-Din al-Tusi al-Abhari Muhyi al-Din al-Maghribi al-Hasan al-Marrakushi Qutb al-Din al-Shirazi Shams al-Din al-Samarqandi Ibn al-Banna' Kamāl al-Dīn al-Fārisī 14th century Nizam al-Din al-Nisapuri Ibn al-Shatir Ibn al-Durayhim Al-Khalili al-Umawi 15th century Ibn al-Majdi al-Rūmī al-Kāshī Ulugh Beg Ali Qushji al-Wafa'i al-Qalaṣādī Sibt al-Maridini Ibn Ghazi al-Miknasi 16th century Al-Birjandi Muhammad Baqir Yazdi Taqi ad-Din Ibn Hamza al-Maghribi
Mathematical works	The Compendious Book on Calculation by Completion and Balancing De Gradibus Principles of Hindu Reckoning Book of Optics The Book of Healing Book on the Measurement of Plane and Spherical Figures Encyclopedia of the Brethren of Purity Toledan Tables Tabula Rogeriana Zij
Concepts	Alhazen's problem Islamic geometric patterns
Centers	Al-Azhar University Al-Mustansiriya University House of Knowledge House of Wisdom Constantinople observatory of Taqi ad-Din Madrasa Maragheh observatory University of al-Qarawiyyin
Influences	Babylonian mathematics Greek mathematics Indian mathematics
Influenced	Byzantine mathematics European mathematics Indian mathematics
Related	Hindu–Arabic numeral system Arabic numerals (Eastern Arabic numerals, Western Arabic numerals) Trigonometric functions History of trigonometry History of algebra

Template:Persondata

• 953 births
• 1029 deaths
• 10th-century mathematicians
• 11th-century mathematicians
• Mathematicians of medieval Islam
• Persian mathematicians
• People from Karaj
• 10th-century Iranian people