Al Karaji enjoyed fame as a mathematician and engineer in his lifetime. Born around 980, he died in 1030. Some dispute exists as to whether his actual name was al-Karaji or al-Karkhi. If his name was al-Karanji, then he is likely from the city of Karaj in Iran. However, if his name was al-Karkhi, then Karkh is a suburb of Baghdad and he likely came from Iraq. The consensus among scholars leans in favor of al-Karaji as the correct name, which would make him a Persian from Iran. His full name may have been Abu Bakr ibn Muhammad ibn al-Husayn al-Karaji.
In any event, we do know that al-Karaji lived in Baghdad the bulk of his life and that he published him most famous mathematical works while under the protection and sponsorship of the rule of Baghdad. In his later life, he retired from the study of mathematics and left Baghdad. He the focused on the study of engineering and published works in that subject.
Underrated historically, many scholars now believe that al-Karaji may have been the first mathematician to introduce the theory of algebraic calculus. Historically, his work introduced the idea of working with unknowns using the tools of arithmetic to solve for unknown. Al-Karaji used mathematical induction in his arguments and made major strides (without reaching) toward understanding inductive proofs.
In addition to making major contributions toward algebra, al-Karaji contributed several ideas in the field of geometry. In his book "On measurement and balances for measuring of buildings and structures," he defines points, lines, surfaces, solids, and angles. He also delves into mapping out geometrical rules for plane and solid figures.
In his work as an official under the ruler of Baghdad, he composed a book that he called his "Sufficient," which was a textbook for civil servants in Baghdad. It instructed them into the use of calculating with integers and fractions, extracting square roots, and determining areas and volumes. He also wrote a small book on basic algebra. He titled his more advanced text on algebra as "The Glorious and the Wonderful." After he left Baghdad, he wrote an engineering book on drilling wells and building aqueducts.
Al-Karaji built on the mathematical theories of earlier mathematicians. In addition, in later generations, those who gave serious study to mathematics relied on the theories of al-Karaji. Thus, in our present generation, al-Karaji is an important link in the chain that leads to our current understanding of algebra and geometry.
History of the Arabs (Tenth Edition), Phillip K. Hitti
Encyclopedia Britannica Online
Al-Karaji entry in Wikipedia