AlKaraji
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Abū Bakr ibn Muḥammad ibn al Ḥusayn alKarajī (or alKarkhī) (c. 953 in Karaj or Karkh – c. 1029) was a 10th century Persian^{[1]} Muslim mathematician and engineer. His three major works are AlBadi' fi'lhisab (Wonderful on calculation), AlFakhri fi'ljabr wa'lmuqabala (Glorious on algebra), and AlKafi fi'lhisab (Sufficient on calculation).
Because most of alKaraji's original works in Arabic are lost, it is not certain what his exact name was. It could either have been alKarkhī, indicating that he was born in Karkh, a suburb of Baghdad, or alKarajī, 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.
ContributionsEdit
AlKaraji was an engineer and mathematician of the highest calibre. AlKaraji wrote on mathematics and engineering.^{[2]} Most regard him as original, in particular for the beginnings of freeing algebra from geometry.
Binomial coefficientsEdit
His enduring contributions to the field of mathematics and engineering are still recognized today in the form of the table of binomial coefficients, its formation law:
and the expansion:
for integer n.
Pure algebra and algebraic calculusEdit
AlKaraji 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 Greek arithmetic) and replace them with the type of arithmetic operations which are at the core of algebra today. His work on algebra and polynomials, gave the rules for arithmetic operations to manipulate polynomials. The historian of mathematics, F. Woepcke, in Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi (Paris, 1853), praised AlKaraji for being "the first who introduced the theory of algebraic calculus". Stemming from this, AlKaraji investigated binomial coefficients and Pascal's triangle.^{[3]}
Algebra of exponentsEdit
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 squarecube, the numerical property of adding exponents was not clear.^{[4]}
Polynomials and monomialsEdit
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.
Proof by inductionEdit
In a now lost work known only from subsequent quotation by alSamaw'al, AlKaraji introduced the idea of proof by mathematical induction. According to mathematics historian Victor J. Katz:
Another important idea introduced by alKaraji and continued by alSamaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus alKaraji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] AlKaraji 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. [...] AlKaraji'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^{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, alKaraji'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 alFakhri is the earliest extant proof of the sum formula for integral cubes.^{[5]}
Integral calculus, binomial theorem, Pascal's triangle, Waring's problemEdit
He was the first to use the method of proof by mathematical induction to prove his results, which he also used to prove the sum formula for integral cubes, an important result in integral calculus.^{[6]} He also used a proof by mathematical induction to prove his discoveries of the general binomial theorem and Pascal's triangle.^{[7]} These were the earliest proofs by mathematical induction for arithmetic sequences, which he introduced in his AlFakhri around 1000 AD. The sum formula for integral cubes is the (true) proposition that every integer can be expressed by the sum of cubed natural numbers. It is a particular case of what is referred to as Waring's problem.^{[8]} His proof was the first to make use of the two basic components of an inductive proof. First, he notes the truth of the statement for n = 1. That is, 1 is the sum of a single cube because 1 = 1^{3}. Secondly, he derives the truth for n = k from that of n = k − 1. For example, when n = 2, it is true that 2 = 1^{3} + 1^{3}. When n = 3, it is true that 3 = 1^{3} + 1^{3} + 1^{3}. The truth of the statement can be extrapolated in this way without limit. Of course, as n grows larger, some of the sums of 1^{3} can be rewritten as the cubes of other natural numbers: for example when n=8 then 8 = 2^{3} = [1^{3} × 8]. Of course, this second component is not explicit since, in some sense, alKaraji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward."^{[9]}^{[10]}
Congruent numbersEdit
In number theory, AlKaraji discovered congruent numbers. According to mathematics historian Lawrence A. D'Antonio: [1]
Congruent numbers can first be found in various works of classical Islamic mathematics, for example, in alKaraji’s early 11th century text, the alFakhri. Congruent numbers then resurface in the treatise Liber Quadratorum of Fibonacci. We then ﬁnd congruent numbers in the inﬂuential 17th century work, Khulasat alHisab of Baha alDin.
See alsoEdit
NotesEdit
 ↑ Classics In The History Of Greek Mathematics  by Jean Christianidis  Page 260
 ↑ http://wwwhistory.mcs.stand.ac.uk/history/Biographies/AlKaraji.html
 ↑ O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn alHusayn AlKaraji", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/AlKaraji.html.
 ↑ Kats, History of Mathematics, first edition, p237
 ↑ Katz (1998), p. 255
 ↑ Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255259. AddisonWesley. ISBN 0321016181.
 ↑ Katz (1998), p. 255:
"Another important idea introduced by alKaraji and continued by alSamaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus alKaraji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] AlKaraji 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. [...] AlKaraji'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^{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, alKaraji'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 alFakhri is the earliest extant proof of the sum formula for integral cubes."
 ↑ Katz (1998), p. 255:
"Another important idea introduced by alKaraji and continued by alSamaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus alKaraji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] AlKaraji 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.
 ↑ Katz (1998), p. 255:
"AlKaraji's argumen includes in essence the two basic components of a modern argument by induction, namely the truth 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, alKaraji'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 alFakhri is the earliest extant proof of the sum formula for integral cubes."
 ↑ O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn alHusayn AlKaraji", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/AlKaraji.html.
References and external linksEdit
 O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn alHusayn AlKaraji", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/AlKaraji.html.
 Rashed, Roshdi (1970–80). "Karajī, Abū Bakr Ibn Muḥammad Ibn al Ḥusayn al". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. ISBN 0684101149.
 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) [PDF]
 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) [PDF]
 "alKaraji" — Encyclopædia Britannica Online (4 April 2006)
