A short proof of the transcendence of the Mahler number

TL;DR

This paper provides concise, accessible proofs demonstrating the transcendence of the Mahler number, complementing existing proofs and making the topic approachable for high-school students.

Contribution

It offers a short, understandable proof of the Mahler number's transcendence, which was previously less accessible and only appeared in recent literature.

Findings

Proofs of transcendence are simplified and made accessible.

The Mahler number's transcendence is established with a concise argument.

The approach is suitable for educational purposes at high-school level.

Abstract

We present short proofs of the transcendence of the Liouville and the Mahler numbers. The first proof is known for a long time, the second proof apparently appeared only in 2002-2003. The proofs are accessible to high-school students.