6Real numbers
IA Numbers and Sets
6.5 Euler’s number
Definition (Euler’s number).
e =
∞
X
j=0
1
j!
= 1 +
1
1!
+
1
2!
+
1
3!
+ ···
This sum exists because the partial sums are bounded by 1+
1
1
+
1
2
+
1
4
+
1
8
···
=
3 and it is increasing. So 2 < e < 3.
Proposition. e is irrational.
Proof.
Is
e ∈ Q
? Suppose
e
=
p
q
. We know
q ≥
2 since
e
is not an integer (it is
between 2 and 3). Then q!e ∈ N. But
q!e = q! + q! +
q!
2!
+
q!
3!
+ ··· +
q!
q!
| {z }
n
+
q!
(q + 1)!
+
q!
(q + 2)!
+ ···
| {z }
x
,
where n ∈ N. We also have
x =
1
q + 1
+
1
(q + 1)(q + 2)
+ ··· .
We can bound it by
0 < x <
1
q + 1
+
1
(q + 1)
2
+
1
(q + 1)
3
+ ··· =
1
q + 1
·
1
1 − 1/(q + 1)
=
1
q
< 1.
This is a contradiction since
q
!
e
must be in
N
but it is a sum of an integer
n
plus a non-integer x.