The series expansion of the number

*e*can be used to prove that

*e*is irrational.

Suppose that *e* = *a*/*b*, for some positive integers *a* and *b*. If we multiply each side of the series expansion

*b*!, we obtain

*b*> 1, this means the entire right side of the original equation cannot be an integer. But this a contradiction, for

*b*!

*e*=

*a*(

*b*-1)! is clearly an integer. This completes the proof.