Liouville number

In number theory, a Liouville number is a real number

x

{\displaystyle x}

with the property that, for every positive integer

n

{\displaystyle n}

, there exists a pair of integers

(

p

,

q

)

{\displaystyle (p,q)}

with

q

>

1

{\displaystyle q>1}

such that

0

<

|

x

−

p

q

|

<

1

q

n

.

{\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{n}}}.}

The inequality implies that Liouville numbers possess an excellent sequence of rational number approximations. In 1844, Joseph Liouville proved a bound showing that there is a limit to how well algebraic numbers can be approximated by rational numbers, and he defined Liouville numbers specifically so that they would have rational approximations better than the ones allowed by this bound. Liouville also exhibited examples of Liouville numbers thereby establishing the existence of transcendental numbers for the first time.

One of these examples is Liouville's constant

L

=

0.110001000000000000000001

…

,

{\displaystyle L=0.110001000000000000000001\ldots ,}

in which the

n

{\displaystyle n}

th digit after the decimal point is 1 if

n

{\displaystyle n}

is the factorial of a positive integer and 0 otherwise. It is known that π and e, although transcendental, are not Liouville numbers.

The existence of Liouville numbers (Liouville's constant)

Liouville numbers can be shown to exist by an explicit construction.

For any integer

b

≥

2

{\displaystyle b\geq 2}

and any sequence of integers

a

1

,

a

2

,

…

{\displaystyle a_{1},a_{2},\dots }

such that

a

k

∈

{

0

,

1

,

2

,

…

,

b

−

1

}

{\displaystyle a_{k}\in \{0,1,2,\ldots ,b-1\}}

for all

k

{\displaystyle k}

and

a

k

≠

0

{\displaystyle a_{k}\neq 0}

for infinitely many

k

{\displaystyle k}

, define the number

x

=

∑

k

=

1

∞

a

k

b

k

!

{\displaystyle x=\sum _{k=1}^{\infty }{\frac {a_{k}}{b^{k!}}}}

.

In the special case when

b

=

10

{\displaystyle b=10}

, and

a

k

=

1

{\displaystyle a_{k}=1}

for all

k

{\displaystyle k}

, the resulting number

x

{\displaystyle x}

is called Liouville's constant:

L

=

0.