Irrational number number to ratios of continuous magnitude. In his commentary on Book 10 of the "Elements", the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows: In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational
Irrational number Consider ; if this is rational, then take "a" = "b" = . Otherwise, take "a" to be the irrational number and "b" = . Then "a" = () = = = 2, which is rational. Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that is transcendental, hence irrational. This theorem states that if "a" and "b" are both algebraic numbers, and "a" is not equal to 0 or 1, and "b" is not a rational number, then any value of "a" is a transcendental number (there can be more than one value