 In the 10th century CE, the Iraqi mathematician Al-Hashimi worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of irrational numbers.An inductive proof for arithmetic sequences was introduced in the "Al-Fakhri" (1000) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle.Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate.