A Geometric Proof that Is Irrational and a New Measure of Its Irrationality

1. INTRODUCTION. While there exist geometric proofs of irrationality for √2 [2], [27], no such proof for e, π , or ln 2 seems to be known. In section 2 we use a geometric construction to prove that e is irrational. The proof leads in section 3 to a new measure of irrationality for e, that is, a lower bound on the distance from e to a given rational number, as a function of its denominator. A connection with the greatest prime factor of a number is discussed in section 4. In section 5 we compare the new irrationality measure for e with a known one, and state a number-theoretic conjecture that implies the known measure is almost always stronger. The new measure is applied in section 6 to prove a special case of a result from [24], leading to another conjecture. Finally, in section 7 we recall a theorem of G. Cantor that can be proved by a similar construction.