e^r is irrational for r∈ℚ∖{0}

e

r

is irrational for

r

∈

ℚ

∖

{

0

}

We here present a proof of the following

theorem

:

Theorem

.

e

r

is

irrational

for all

r

∈

Q

∖

{

0

}

To begin with, note that it is

sufficient

to show that

e

u

is irrational for any

positive integer

(

http://planetmath.org/NaturalNumber

)

1

1

In this entry,

ℕ

:=

{

1

,

2

,

3

,

…

}

and

ℕ

0

:=

ℕ

∪

{

0

}

.

u

(for if

e

r

=

e

u

v

were

rational

, so would

(

e

u

v

)

v

=

e

u

). Next, we look at some simple properties of

polynomial

f

n

⁢

(

x

)

:=

x

n

⁢

(

1

-

x

)

n

n

!

:

•

f

n

⁢

(

x

)

=

1

n

!

⁢

∑

i

=

n

2

⁢

n

c

i

⁢

x

i

, with

c

i

∈

ℤ

for all

i

.

•

f

n

(

k

)

⁢

(

0

)

and

f

n

(

k

)

⁢

(

1

)

are

integers

for all

k

∈

ℕ

0

: as

0

is a root (

http://planetmath.org/Root

) of order

n

,

f

n

(

k

)

⁢

(

0

)

=

0

unless

n

≤

k

≤

2

⁢

n

, in which case

f

n

(

k

)

⁢

(

0

)

=

k

!

n

!

⁢

c

k

, an integer. Since

f

n

(

k

)

⁢

(

x

)

=

(

-

1

)

k

⁢

f

n

(

k

)

⁢

(

1

-

x

)

, the same is true for

f

n

(

k

)

⁢

(

1

)

.

•

For all

0

<

x

<

1

we have

0

<

f

n

⁢

(

x

)

<

1

n

!

.

Now we can readily prove the theorem:

Proof.

Assume that

e

u

=

a

b

for some

(

a

,

b

)

∈

ℕ

2

and let

F

n

⁢

(

x

)

:=

∑

k

=

0

∞

(

-

1

)

k

⁢

u

2

⁢

n

-

k

⁢

f

n

(

k

)

⁢

(

x

)

,

which is actually a finite sum since

f

n

(

k

)

⁢

(

x

)

=

0

for all

k

>

2

⁢

n

. Differentiating

F

n

⁢

(

x

)

yields

F

n

′

⁢

(

x

)

=

u

2

⁢

n

+

1

⁢

f

n

⁢

(

x

)

-

u

⁢

F

n

⁢

(

x

)

and thus:

d

d

⁢

x

⁢

[

e

u

⁢

x

⁢

F

n

⁢

(

x

)

]

=

u

⁢

e

u

⁢

x

⁢

F

n

⁢

(

x

)

+

e

u

⁢

x

⁢

F

n

′

⁢

(

x

)

=

u

2

⁢

n

+

1

⁢

e

u

⁢

x

⁢

f

n

⁢

(

x

)

.

Now consider the

sequence

(

w

n

)

n

∈

ℕ

:=

b

⁢

∫

0

1

u

2

⁢

n

+

1

⁢

e

u

⁢

x

⁢

f

n

⁢

(

x

)

⁢

𝑑

x

=

b

⁢

[

e

u

⁢

x

⁢

F

n

⁢

(

x

)

]

0

1

=

a

⁢

F

n

⁢

(

1

)

-

b

⁢

F

n

⁢

(

0

)

.

Given the remarks on

f

n

⁢

(

x

)

,

w

n

should be an integer for all

n

∈

ℕ

, yet it is clear that

w

n

<

b

⁢

u

2

⁢

n

+

1

⁢

1

n

!

=

a

n

!

⁢

u

2

⁢

n

+

1

and so

lim

n

→

∞

⁡

w

n

=

0

, a

contradiction

.
∎

The result could also easily have been obtained by starting with

w

n

and integrating by parts

2

⁢

n

times. Note also that much stronger statements are known, such as “

e

a

is transcendental for all

a

∈

𝔸

∖

{

0

}

”

2

2

𝔸

denotes the set of

algebraic numbers

.

. We conclude this entry with the following evident corollary:

Corollary

.

For all

r

∈

Q

+

,

log

⁡

r

is irrational.

References

1

M. Aigner & G. M. Ziegler

:

Proofs from THE BOOK

, 3

rd

edition (2004), Springer-Verlag, 30–31.

2

G. H. Hardy & E. M. Wright

:

An

Introduction

to the Theory of Numbers

, 5

th

edition (1979), Oxford University Press, 46–47.

Title

e

r

is irrational for

r

∈

ℚ

∖

{

0

}

Canonical name

ErIsIrrationalForRinmathbbQsetminus0

Date of creation

2013-03-22 15:07:46

Last modified on

2013-03-22 15:07:46

Owner

Cosmin (8605)

Last modified by

Cosmin (8605)

Numerical id

12

Author

Cosmin (8605)

Entry type

Theorem

Classification

msc 11J72

Synonym

e

r

is irrational for non-zero rational r

Synonym

irrationality of the

exponential function

on

ℚ

Related topic

Irrational

Related topic

EIsIrrationalProof

Related topic

EIsIrrational

Related topic

EIsTranscendental

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