Proof that e is irrational


Presume for contradiction that e is rational, therefore

Such that p and q are integers. Since q is an integer there must be the term  in the series of e, thus

Multiply both sides by q to get

The term p(q-1)! must be an integer as is   since q! is divisible by all factorials including q!.

It follows that

The left side is greater than zero. The right side

is an infinite series with


demonstrating that  is not an integer. Thus

 for integer p and q.  Therefore e is irrational. Q.E.D.