Proof that e is transcendental

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.