Proof that e is irrational
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
-
by
b!, we obtain
-
The first term on the right side of this equation is an integer. The remainder of the right side is a positive number bounded above by the
geometric series
-
Since
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.
∎