The first proof that the base of the natural logarithms, e, is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following: Assume, for purpose of finding a contradiction, that e is algebraic. Then there exists a finite set of integer coefficients satisfying the equation: and such that and are both non-zero. We have arrived at the equation: which can now be written in the form where is a non-zero integer and is not. To show that for sufficiently large k for every real number G