Among historians, his most widely studied work is his algebra book al-fakhri fi al-jabr wa al-muqabala, which survives from the medieval era in at least four copies.

[6] He expounded the basic principles of hydrology[7] and this book reveals his profound knowledge of this science and has been described as the oldest extant text in this field.

[8][9][10] He systematically studied the algebra of exponents, and was the first to define the rules for monomials like x,x²,x³ and their reciprocals in the cases of multiplication and division.

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.

[17] As Katz says 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.