# 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 $c_{0},c_{1},\ldots,c_{n},$ satisfying the equation:

$c_{0}+c_{1}e+c_{2}e^{2}+\cdots+c_{n}e^{n}=0$

and such that $c_0$ and $c_n$ 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 $\int^{\infty}_{0}$, where the notation $\int^{b}_{a}$ will be used in this proof as shorthand for the integral:

$\int^{b}_{a}:=\int^{b}_{a}x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x}\,dx.$

We have arrived at the equation:

$c_{0}\int^{\infty}_{0}+c_{1}e\int^{\infty}_{0}+\cdots+c_{n}e^{n}\int^{\infty}_{0} = 0$

which can now be written in the form

$P_{1}+P_{2}=0\;$

where

$P_{1}=c_{0}\int^{\infty}_{0}+c_{1}e\int^{\infty}_{1}+c_{2}e^{2}\int^{\infty}_{2}+\cdots+c_{n}e^{n}\int^{\infty}_{n}$
$P_{2}=c_{1}e\int^{1}_{0}+c_{2}e^{2}\int^{2}_{0}+\cdots+c_{n}e^{n}\int^{n}_{0}$

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

$\frac{P_{1}}{k!}$ is a non-zero integer and $\frac{P_{2}}{k!}$ is not.

The fact that $\frac{P_{1}}{k!}$ is a nonzero integer results from the relation

$\int^{\infty}_{0}x^{j}e^{-x}\,dx=j!$

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

To show that

$\left|\frac{P_{2}}{k!}\right|<1$ for sufficiently large k

we first note that $x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x}$ is the product of the functions $[x(x-1)(x-2)\cdots(x-n)]^{k}$ and $(x-1)(x-2)\cdots(x-n)e^{-x}$. Using upper bounds for $|x(x-1)(x-2)\cdots(x-n)|$ and $|(x-1)(x-2)\cdots(x-n)e^{-x}|$ on the interval [0,n] and employing the fact

$\lim_{k\to\infty}\frac{G^k}{k!}=0$ 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.