e and pi are transcendental

Nevertheless there are uncountably infinite transcendental numbers, we know few of them. This is because they do not have a closed structure. Consequently the heuristic method is required to study them.

1737	L. Euler	proved that $e$ and $e^2$ are irrational.
1761	J. Lambert	proved that $\pi$ is irrational.
1840	J. Liouville	proved that $e$ and $e^2$ are not algebraic numbers of degree 2.
1844	J. Liouville	constructed an artificial transcendental number.
1873	C. Hermite	proved that $e$ is transcendental.
1874	G. Cantor	proved that almost all numbers are transcendental.
1882	F. Lindemann	proved that $\pi$ is transcendental by generalizing Hermite's method.
1900	D. Hilbert	outlined 23 major mathematical problems for the 20th century.
		The 7th problem is that the expression $\alpha^\beta$, for an algebraic base $\alpha$ and an irrational algebraic exponent $\beta$, e. g., the number $2^{\sqrt{2}}$ or $e^{\pi}= (-1)^{-i}$, always represents a transcendental or at least an irrational number.
1929	A. Gel'fond	proved that $2^{\sqrt{-2}}$ is transcendental.
1929	C. Siegel	proved that $2^{\sqrt{2}}$ is transcendental by using Gel'fond's result and established a new method for the algebraic independence of values of certain E-functions.
1934	A. Gel'fond	solved Hilbert's 7th problem affirmatively.
1935	T. Schneider	also solved it independently.
1966	A. Baker	proved that the expression $\alpha \log \beta$, for algebraic numbers $\alpha$ and $\beta$, is transcendental.

This article introduces transcendence of $e$ and $\pi$ according to A. Baker's method (reductio ad absurdum).

Preparation