- 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 )

*Wikimedia Foundation.
2010.*

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**karaji**— var. koradji … Useful english dictionary**Karajī, al-**— ▪ Persian mathematician and engineer 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… … Universalium**Al-Karaji**— Abu Bekr Muhammad ibn al Hasan al Karaji (árabe: Abu Bakr Muhammad ibn al Hasan Alkrgi) (c. 953 – c. 1029), fue un matemático e ingeniero persa. Vivió y trabajó la mayor parte de su vida en Bagdad, que era la capital científica y… … Wikipedia Español**Al-Karaji**— Abu Bakr Muhammad ibn al Hasan al Karaji (en arabe : أبو بكر محمد بن الحاسب الكرجي), aussi connu sous le nom de Al karkhi, né vers 953, mort vers 1029 à Karaj, était un mathématicien et ingénieur perse musulman. Ses travaux principaux sont… … Wikipédia en Français**Karajimiškis**— Karaji̇̀miškis dkt … Bendrinės lietuvių kalbos žodyno antraštynas**Timeline of Islamic science and engineering**— This timeline of Islamic science and engineering covers the general development of science and technology in the Islamic world during the Islamic Golden Age, usually dated from the 7th to 16th centuries.From the 17th century onwards, the advances … Wikipedia**History of algebra**— Elementary algebra is the branch of mathematics that deals with solving for the operands of arithmetic equations. Modern or abstract algebra has its origins as an abstraction of elementary algebra. Historians know that the earliest mathematical… … Wikipedia**Matemática en el Islam medieval**— Saltar a navegación, búsqueda Contenido 1 Valoración de la ciencia islámica 2 Desarrollos y contexto histórico 3 Otros ejemplos de desarrollo … Wikipedia Español**mathematics**— /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… … Universalium**Matemática en el islam medieval**— Tratado de arte numeral de Joannis de Sacro Bosco. La matemática árabe se enriqueció en forma creciente a medida que los musulmanes conquistaron territorios. Con rapidez inusitada, el islamismo se expandió en todo el territorio que se extiende… … Wikipedia Español