An Introduction to Irrationality and Transcendence Methods

One of the most well known proofs is to argue by contradiction as follows: assume √ 2 is rational and write it as a/b where a and b are relatively prime positive rational integers. Then a = 2b. It follows that a is even. Write a = 2a′. From 2a′ = b one deduces that b also is even, contradicting the assumption that a and b were relatively prime. There are variants of this proof a number of them are in the nice booklet [19]. For instance using the relation