- Al-Karaji
**transl|ar|ALA|Abū Bakr ibn Muḥammad ibn al Ḥusayn al-Karajī**(or**transl|ar|ALA|al-Karkhī**) (c. 953 inKaraj orKarkh – c. 1029) was a 10th century Persian [*Classics In The History Of Greek Mathematics - by Jean Christianidis - Page 260*] Muslim mathematician and engineer. His three major works are "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").Because al-Karaji's original works in

Arabic are lost, it is not certain what his exact name was. It could either have been "al-Karkhī", indicating that he was born inKarkh , a suburb ofBaghdad , or "al-Karajī" indicating his family came from the city ofKaraj . He certainly lived and worked for most of his life in Baghdad, however, which was the scientific and trade capital of theIslam ic world.Al-Karaji was an engineer and mathematician of the highest calibre. His enduring contributions to the field of mathematics and engineering are still recognized today in the form of the table of

binomial coefficient s, its formation law::$\{n\; choose\; m\}\; =\; \{n-1\; choose\; m-1\}\; +\; \{n-1\; choose\; m\}$

and the expansion:

:$(a+b)^n=sum\_\{k=0\}^n\{n\; choose\; k\}a^kb^\{n-k\}$

for integer n.

Al-Karaji wrote about the work of earlier mathematicians, and he is now regarded as the first person to free

algebra from geometrical operations, that were the product of Greekarithmetic , and replace them with the type of operations which are at the core of algebra today. His work onalgebra andpolynomial s, gave the rules for arithmetic operations to manipulate polynomials. Thehistorian of mathematics, F. Woepcke, in "Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi" (Paris , 1853), praised Al-Karaji for being "the first who introduced thetheory ofalgebra iccalculus ". Stemming from this, Al-Karaji investigatedbinomial coefficients andPascal's triangle . [*MacTutor|id=Al-Karaji|title=Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji*]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 forintegral cubes, an important result inintegral calculus . [*Victor J. Katz (1998). "History of Mathematics: An Introduction", p. 255-259.*] He also used a proof by mathematical induction to prove theAddison-Wesley . ISBN 0321016181.binomial theorem andPascal's triangle . [*Katz (1998), p. 255:*

]*"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 thetruth 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."**ee also***

Islamic science

*List of Iranian scientists **Notes****References and external links***

*

* J. Christianidis. "Classics in the History of Greek Mathematics", p. 260

* Carl R. Seaquist, Padmanabhan Seshaiyer, and Dianne Crowley. [*http://www.cs.southwestern.edu/txcmj/aculture4f.pdf "Calculation across Cultures and History"*] ("Texas College Mathematics Journal" 1:1, 2005; pp 15–31) [PDF]

* Matthew Hubbard and Tom Roby. [*http://binomial.csuhayward.edu/MidEast.html "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 . [*http://math.berkeley.edu/~wodzicki/160/HistIntr.pdf "Early History of Algebra: a Sketch"*] ("Math" 160, Fall 2005) [PDF]

* [*http://0-www.search.eb.com.library.uor.edu/eb/article-9343818 "al-Karaji"*] — "Encyclopædia Britannica" Online (4 April 2006 )

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