**Al-Karaji** was an Islamic mathematician who wrote about the work of earlier mathematicians and who can be regarded as the first person to free algebra from geometrical operations and replace them with the type of operations which are at the core of algebra today.

- It appears both as al-Karaji and as al-Karkhi but this is not a simple matter of two different transliterations of the same Arabic name.
- Certainly we know that al-Karaji lived in Baghdad for most of his life and that his chief mathematical works were written during the time when he lived in that city.
- However, at some later point in his career, al-Karaji left Baghdad to live in what are described as the "mountain countries".
- The importance of al-Karaji in the development of mathematics is viewed rather differently by different authors.
- al-Karaji usually gives a numerical example for his rules but does not give any sort of proof beyond giving geometrical pictures.
- the discovery and reading of the arithmetical work of Diophantus, in the light of the algebraic conceptions and methods of al-Khwarizmi and other Arab algebraists, made possible a new departure in algebra by Al-Karaji ...
- Having given rules for multiplication and division of monomials al-Karaji then looked at "composite quantities" or sums of monomials.
- Al-Karaji also uses a form of mathematical induction in his arguments, although he certainly does not give a rigorous exposition of the principle.
- One of the results on which al-Karaji uses this form of induction comes from his work on the binomial theorem, the binomial coefficients and the Pascal triangle.
- The general construction of the Pascal triangle was given by al-Karaji in work described in the later writings of al-Samawal.
- Al-Karaji said that in order to succeed we must place 'one' on a table and 'one' below the first 'one', move the first 'one' into a second column, add the first 'one' to the 'one' below it.
- The table al-Karaji constructed looks like the Pascal triangle on its side.
- Other results obtained by al-Karaji include summing the first nnn natural numbers, the squares of the first nnn natural numbers and the cubes of these numbers.
- Al-Karaji showed that (1+2+3+...+10)2(1 + 2 + 3 + ...
- Finally we should mention the influence of Diophantus on al-Karaji.
- The first five books of Diophantus's Arithmetica had been translated into Arabic by ibn Liqa around 870 and these were studied by al-Karaji.
- more than a third of the problems of the first book of Diophantus, the problems of the second book starting with the eighth, and virtually all the problems of the third book were included by al-Karaji in his collection.
- Al-Karaji also invented many new problem of his own but even those of Diophantus were certainly not just taken without further development.
- It was not only to algebra that al-Karaji contributed.
- al-Karaji defines points, lines, surfaces, solids and angles.

Born 13 April 953, Baghdad (now in Iraq). Died about 1029.

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Ancient Arab, Origin Iraq, Special Numbers And Numerals

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive