The well-known simple proof that the number e is irrational consists of writing e as its exponential series, multiplying by a suitable factorial, r!, and deriving the contradiction that, if e were rational, there would be an integer lying strictly between 0 and 1. This proof is referred to as a ‘trick’ in the recent highly acclaimed book Proofs from THE BOOK [1, p. 27], where the surprising statement is made that ‘This trick … isn’t even good enough to prove that e2 is irrational’. In this article we show that there are nevertheless two similar simple proofs that e2 is irrational. The first proof is in fact well known [2, p. 12]. The second, though elementary, is less obvious and certainly not well known but, together with the idea in the first proof, it implies the even stronger property that e4 is irrational.