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:
and the expansion:
for integer n.
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.
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. He also used a proof by mathematical induction to prove the binomial theorem and Pascal's triangle.