An introduction to transcendental numbers

After a brief introduction to the algebraic numbers, we turn to the definition and history of the transcendental numbers. In 1844, Joseph Liouville proved the existence of transcendental numbers by approximating reals by rationals, and provided examples of such numbers. Georg Cantor produced a second proof of the existence of transcendental numbers in 1874 using infinite cardinal numbers, but provided no means of finding them. Charles Hermite proved that e was transcendental in 1873. His proof was later simplified by Karl Weierstrass, David Hilbert, Adolf Hurwitz, and Paul Gordan. This simplified proof will be presented.