Formalizing a Proof that e is Transcendental

A transcendental number is one that is not the root of any non-zero polynomial having integer coefficients. It immediately follows that no rational number q is transcendental, since q can be written as a/b where a and b are integers, and thus q is a root of bx − a. Furthermore, the transcendentals are a proper subset of the irrationals, since for example the irrational √ 2 is a root of x − 2. The existence of transcendental numbers was first established by Liouville in 1844 [11] by exhibiting a transcendental continued fraction. A later and simpler proof for their existence is due to Cantor [2, 4], who used a straightforward counting argument to show that the non-transcendental (called algebraic) numbers are countable. The result then follows from the fact that the reals are uncountable. The first decimal number demonstrated to be transcendental ∑∞ n=1 10 −n! has come to be known as Liouville’s constant. Transcendental numbers play an important role in Mathematics historically; for examples the fact that π is transcendental was used in the proof of the impossibility of squaring the circle, and the 7th Hilbert problem pertains to transcendental numbers. The first non-fabricated number proven to be transcendental was the base of the natural logarithm e, as established by Hermite in 1873 [9]. This paper describes a formalization of (a simplification of) Hermite’s proof using the HOL Light theorem prover; this is the first time this theorem has been formalized.