About the irrationality of e=2.718...
We prove that e, basis of the neperien logarithm, is not a rational number.
The same method can also be used to prove that e cannot be a root of a second
degree equation with rational coeﬃcients, i.e. we cannot ﬁnd ae +
that a is a positive integer and b, c are positive or negati ve integers. As a matter
of fact, if we replace e and 1/e or e
with their power series deduced from e
and given that we multiply both sides of the equation by 1 ⇥ 2 ⇥ 3 ⇥ ... ⇥ n,we
where µ is an integer. We can make it so that
is positive; we just need to suppose that n is even when b<0 and t h at n is
odd when b>0; for big values of n the equation above leads to a contradiction;
the ﬁrst term of the equation is positive and very small, with values between 0
and 1, and as such can never be equal to an integer µ. So, etc.