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 A stronger result is the following: Every rational number in the interval formula_23 can be written either as "a" for some irrational number "a" or as "n" for some natural number "n". Similarly, every positive rational number can be written either as formula_24 for some irrational number "a" or as formula_25 for some natural number "n". It is not known if formula_26 (or formula_27) is irrational. In fact, there is no pair of non-zero integers formula_28 for which it is known whether formula_29 is irrational. Moreover, it is not known whether or not the set formula_30 is algebraically independent over