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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[1] 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 most of 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 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.

## ContributionsEdit

Al-Karaji was an engineer and mathematician of the highest calibre. Al-Karaji 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:

${n \choose m} = {n-1 \choose m-1} + {n-1 \choose m}$

and the expansion:

$(a+b)^n=\sum_{k=0}^n{n \choose k}a^kb^{n-k}$

for integer n.

### Pure algebra and algebraic calculusEdit

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 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 Al-Karaji for being "the first who introduced the theory of algebraic calculus". Stemming from this, Al-Karaji 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 square-cube, 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 al-Samaw'al, Al-Karaji introduced the idea of proof by mathematical induction. According to mathematics historian Victor J. Katz:

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 = 13) 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.[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 Al-Fakhri 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 = 13. Secondly, he derives the truth for n = k from that of nk − 1. For example, when n = 2, it is true that 2 = 13 + 13. When n = 3, it is true that 3 = 13 + 13 + 13. The truth of the statement can be extrapolated in this way without limit. Of course, as n grows larger, some of the sums of 13 can be rewritten as the cubes of other natural numbers: for example when n=8 then 8 = 23 = [13 × 8]. 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."[9][10]

### Congruent numbersEdit

In number theory, Al-Karaji 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 al-Karaji’s early 11th century text, the al-Fakhri. Congruent numbers then resurface in the treatise Liber Quadratorum of Fibonacci. We then ﬁnd congruent numbers in the inﬂuential 17th century work, Khulasat al-Hisab of Baha al-Din.

## NotesEdit

1. Classics In The History Of Greek Mathematics - by Jean Christianidis - Page 260
2. http://www-history.mcs.st-and.ac.uk/history/Biographies/Al-Karaji.html
3. Kats, History of Mathematics, first edition, p237
4. Katz (1998), p. 255
5. Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255-259. Addison-Wesley. ISBN 0321016181.
6. 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 the truth of the statement for n = 1 (1 = 13) 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."
7. 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.
8. Katz (1998), p. 255:
"Al-Karaji'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 = 13) 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."