# Proof that e is transcendental

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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.

Depending on the value of *n*, we specify a sufficiently large positive integer *k* (to meet our needs later), and multiply both sides of the above equation by , where the notation will be used in this proof as shorthand for the integral:

We have arrived at the equation:

which can now be written in the form

where

The plan of attack now is to show that for *k* sufficiently large, the above relations are impossible to satisfy because

- is a non-zero integer and is not.

The fact that is a nonzero integer results from the relation

which is valid for any positive integer *j* and can be proved using integration by parts and mathematical induction.

To show that

- for sufficiently large
*k*

we first note that
is the product of the functions and .
Using upper bounds for and on the interval [0,*n*] and employing the fact

- for every real number
*G*

is then sufficient to finish the proof.

A similar strategy, different from Lindemann's original approach, can be used to show that the number *π* is transcendental. Besides the gamma-function and some estimates as in the proof for *e*, facts about symmetric polynomials play a vital role in the proof.