# Al-Karaji: Wikis

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# Encyclopedia

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 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.

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 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.

He was the first who introduced the theory of algebraic calculus acorrding to F. Woepcke, in Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi (Paris, 1853) 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 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.[2]

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 for integral cubes, an important result in integral calculus.[3] He also used a proof by mathematical induction to prove the binomial theorem and Pascal's triangle.[4]

## Notes

1. ^ Classics In The History Of Greek Mathematics - by Jean Christianidis - Page 260
2. ^  .
3. ^ Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255-259. Addison-Wesley. ISBN 0321016181.
4. ^ 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."