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Joseph Liouville (1809 – 1882) was one of the leading French mat...
Note that Liouville doesn't use the factorial symbol "!". Although ...
If we multiply both sides by $n!$ and we use $e^{x}= \sum_{l=0}^{\i... Liouville proves that$e\$ is not an algebraic number of degree 2 (i...
This is the original paper published in the French journal "Journal...
J. Liouville
1840
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 coecients, i.e. we cannot ﬁnd ae +
b
e
= c,such
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
1
with their power series deduced from e
x
,
and given that we multiply both sides of the equation by 1 2 3 ... n,we
ﬁnd that
a
n +1
1+
1
n +2
+ ...
±
b
n +1
1
1
n +2
+ ...
= µ
where µ is an integer. We can make it so that
±
b
n +1
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.
1