( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century

Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples ** Persian language, an Iranian language of the ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

and

engineer Engineers, as practitioners of engineering, are professionals who invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the l ...

who flourished at

Baghdad Baghdad (; ar, بَغْدَاد , ) is the capital of Iraq and the second-largest city in the Arab world after Cairo. It is located on the Tigris near the ruins of the ancient city of Babylon and the Sassanid Persian capital of Ctesiphon ...

. He was born in

Karaj Karaj ( fa, کرج, ) is the capital of Alborz Province, Iran, and effectively a satellite city of Tehran. Although the county hosts a population around 1.97 million, as recorded in the 2016 census, most of the county is rugged mountain. The urb ...

, a city near

Tehran Tehran (; fa, تهران ) is the largest city in Tehran Province and the capital of Iran. With a population of around 9 million in the city and around 16 million in the larger metropolitan area of Greater Tehran, Tehran is the most popul ...

. 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'').

Work

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

Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...

) 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 Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...

Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...

, but long after

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...

and

Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...

. He expounded the basic principles of hydrology and this book reveals his profound knowledge of this science and has been described as the oldest extant text in this field. He systematically studied the algebra of exponents, and was the first to define the rules for monomials like x,x^2,x^3 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. His work on

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

and

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

s 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. Al-Karaji gave the first formulation of the

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

s and the first description of

Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...

. He is also credited with the discovery of the binomial theorem. In a now lost work known only from subsequent quotation by

al-Samaw'al Al-Samawʾal ibn Yaḥyā al-Maghribī ( ar, السموأل بن يحيى المغربي, ; c. 1130 – c. 1180), commonly known as Samau'al al-Maghribi, was a mathematician, Islamic astronomy, astronomer and Islamic medicine, physician. Born to ...

Al-Karaji introduced the idea of argument by

mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...

. As Katz says

Notes

References and external links

* * * 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) DF* 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 Mariusz Wodzicki (Count Wodzicki) (born 1956) is a Polish mathematician and nobleman, whose works primarily focus on analysis, algebraic k-theory, noncommutative geometry, and algebraic geometry. Wodzicki was born in Bytom, Poland in 1956. He rec ...

"Early History of Algebra: a Sketch"
(''Math'' 160, Fall 2005) DF
"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. {{DEFAULTSORT:Karaji 950s births 1029 deaths 10th-century Iranian mathematicians 11th-century Iranian mathematicians People from Karaj

Work

See also

Notes

References and external links