no such proof for e ,

While there exist geometric proofs of irrationality for √ 2 [

1. INTRODUCTION. While there exist geometric proofs of irrationality for √ 2 [ 2 ], [ 27 ],

and a New Measure of it s Irrationali ty

seems to be known. In section 2 we use a geometric

construction to prove that e is irrational. (For other proofs , see [ 1 , pp. 27- 28], [ 3 , p. 352],

[ 6 ], [ 10 , pp. 78-79], [ 15 , p. 301], [ 16 ], [ 17 , p. 11], [ 19 ], [ 20 ], and [ 21 , p. 302].) 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.

2. PROOF. The irrationality of e is a consequence of the following construction of a

nested sequence of closed intervals