Ab out the irrationalit y of e=2.718...

J. Liouville

1840

W e pro ve that e , basis of the nep erien logarithm, is not a rational n umber.

The same metho d can also be used to prov e that e cannot b e a root of a second

degree equation with rational coe ﬃ cients, i.e. w e cannot ﬁnd ae +

b

e

= c ,s u c h

that a is a p ositiv e in teger and b , c are positive or negati ve integers. As a matter

of fact, if w e replace e and 1 /e or e

 1

with their p o wer series deduced from e

x

,

and given that w e multiply b oth sides of the equation b y 1 ⇥ 2 ⇥ 3 ⇥ ... ⇥ n ,w e

ﬁnd that

a

n +1

✓

1+

1

n +2

+ ...

◆

±

b

n +1

✓

1 

1

n +2

+ ...

◆

= µ

where µ is an in teger. W e can make it so that

±

b

n +1

is positive; we just need to supp ose that n is ev en when b< 0 and t h at n is

od d w he n b> 0; for big v alues of n the equation ab o ve leads to a con tradiction;

the ﬁrst term of the equation is p ositiv e and very small, with v alues betw een 0

and 1, and as suc h can never b e equal to an in teger µ . So, etc.

1