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.