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
Proof that e is irrational a proof that "e" is not a root of a second degree polynomial with rational coefficients. This last fact implies that "e" is irrational. His proofs are similar to Fourier's proof of the irrationality of "e". In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that "e" is not a root of a third degree polynomial with rational coefficients. In particular, "e" is irrational. More generally, "e" is irrational for any non-zero rational "q". Proof that e is irrational The number "e" was introduced by Jacob Bernoulli in 1683. More than half a