In Egypt, Abu Kamil extended algebra to the set of irrational numbers无理数, accepting square roots and fourth roots as solutions and coefficients to quadratic equations. He also developed techniques used to solve three non-linear非线性的 simultaneous equations联立方程组 with three unknown variables变量. One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found 2676 solutions. His works formed an important foundation for the development of algebra and influenced later mathematicians, such as al-Karaji and Fibonacci斐波那契. Further developments in algebra were made by Al-Karaji in his treatise论文 al-Fakhri, where he extends the methodology方法、原则 to incorporate包含 integer powers整数幂 and integer roots整根 of unknown quantities数、量. Something close to a proof by mathematical induction数学归纳法 appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem二次项定理, Pascal's triangle杨辉三角, and the sum of integral cubes整数立方和.
The historian of mathematics, F. Woepcke, praised Al-Karaji for being "the first who introduced the theory of algebraic calculus代数微积分." Also in the 10th century, Abul Wafa translated the works of Diophantus into Arabic. Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily快捷的 generalizable可推广的 for determining the general formula for the sum of any integral powers. He performed an integration积分 in order to find the volume of a paraboloid抛物体, and was able to generalize概况、归纳 his result for the integrals of polynomials多项式 up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.