Liouville number

In number theory, a Liouville number is a real number ${\displaystyle x}$ with the property that, for every positive integer ${\displaystyle n}$, there exists a pair of integers ${\displaystyle (p,q)}$ with ${\displaystyle q>1}$ such that

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

Liouville numbers are "almost rational", and can thus be approximated "quite closely" by sequences of rational numbers. Precisely, these are transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number can be. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental,^[1] thus establishing the existence of transcendental numbers for the first time.^[2] It is known that π and e are not Liouville numbers.^[3]

The existence of Liouville numbers (Liouville's constant)

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

For any integer ${\displaystyle b\geq 2}$  and any sequence of integers ${\displaystyle (a_{1},a_{2},\dots )}$  such that ${\displaystyle a_{k}\in \{0,1,2,\dots ,b-1\}}$  for all ${\displaystyle k}$  and ${\displaystyle a_{k}\neq 0}$  for infinitely many ${\displaystyle k}$ , define the number

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

 

In the special case when ${\displaystyle b=10}$ , and ${\displaystyle a_{k}=1}$  for all ${\displaystyle k}$ , the resulting number ${\displaystyle x}$  is called Liouville's constant:

L = 0.11000100000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001...

It follows from the definition of ${\displaystyle x}$  that its base-${\displaystyle b}$  representation is

${\displaystyle x=\left(0.a_{1}a_{2}000a_{3}00000000000000000a_{4}0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a_{5}\ldots \right)_{b}\;}$ 

where the ${\displaystyle n}$ th term is in the ${\displaystyle (n!)}$ th place.

Since this base-${\displaystyle b}$  representation is non-repeating it follows that ${\displaystyle x}$  is not a rational number. Therefore, for any rational number ${\displaystyle p/q}$ , ${\displaystyle |x-p/q|>0}$ .

Now, for any integer ${\displaystyle n\geq 1}$ , ${\displaystyle p_{n}}$  and ${\displaystyle q_{n}}$  can be defined as follows:

$${\displaystyle q_{n}=b^{n!}\,;\quad p_{n}=q_{n}\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}=\sum _{k=1}^{n}{a_{k}}{b^{n!-k!}}\;.}$$

 

Then,

$${\displaystyle {\begin{aligned}0<\left|x-{\frac {p_{n}}{q_{n}}}\right|&=\left|x-{\frac {q_{n}\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}}{q_{n}}}\right|=\left|x-\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}\right|=\left|\sum _{k=1}^{\infty }{\frac {a_{k}}{b^{k!}}}-\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}\right|=\left|\left(\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}+\sum _{k=n+1}^{\infty }{\frac {a_{k}}{b^{k!}}}\right)-\sum _{k=1}^{n}{\frac {a_{k}}{b^{k!}}}\right|=\sum _{k=n+1}^{\infty }{\frac {a_{k}}{b^{k!}}}\\[6pt]&\leq \sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}<\sum _{k=(n+1)!}^{\infty }{\frac {b-1}{b^{k}}}={\frac {b-1}{b^{(n+1)!}}}+{\frac {b-1}{b^{(n+1)!+1}}}+{\frac {b-1}{b^{(n+1)!+2}}}+...={\frac {b-1}{b^{(n+1)!}b^{0}}}+{\frac {b-1}{b^{(n+1)!}b^{1}}}+{\frac {b-1}{b^{(n+1)!}b^{2}}}+...={\frac {b-1}{b^{(n+1)!}}}\sum _{k=0}^{\infty }{\frac {1}{b^{k}}}\\[6pt]&={\frac {b-1}{b^{(n+1)!}}}\cdot {\frac {b}{b-1}}={\frac {b}{b^{(n+1)!}}}\leq {\frac {b^{n!}}{b^{(n+1)!}}}={\frac {1}{b^{(n+1)!-n!}}}={\frac {1}{b^{(n+1)n!-n!}}}={\frac {1}{b^{n(n!)+n!-n!}}}={\frac {1}{b^{(n!)n}}}={\frac {1}{{q_{n}}^{n}}}\end{aligned}}}$$

 

Therefore, any such ${\displaystyle x}$  is a Liouville number.

Notes on the proof

1. The inequality ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {a_{k}}{b^{k!}}}\leq \sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}}$  follows since a_k ∈ {0, 1, 2, …, b−1} for all k, so at most a_k = b−1. The largest possible sum would occur if the sequence of integers (a₁, a₂, …) were (b−1, b−1, ...), i.e. a_k = b−1, for all k. ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {a_{k}}{b^{k!}}}}$  will thus be less than or equal to this largest possible sum.
2. The strong inequality ${\displaystyle {\begin{aligned}\sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}<\sum _{k=(n+1)!}^{\infty }{\frac {b-1}{b^{k}}}\end{aligned}}}$  follows from the motivation to eliminate the series by way of reducing it to a series for which a formula is known. In the proof so far, the purpose for introducing the inequality in #1 comes from intuition that ${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{b^{k}}}={\frac {b}{b-1}}}$  (the geometric series formula); therefore, if an inequality can be found from ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {a_{k}}{b^{k!}}}}$  that introduces a series with (b−1) in the numerator, and if the denominator term can be further reduced from ${\displaystyle b^{k!}}$ to ${\displaystyle b^{k}}$ , as well as shifting the series indices from 0 to ${\displaystyle \infty }$ , then both series and (b−1) terms will be eliminated, getting closer to a fraction of the form ${\displaystyle {\frac {1}{b^{{\text{exponent}}\times n}}}}$ , which is the end-goal of the proof. This motivation is increased here by selecting now from the sum ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}}$  a partial sum. Observe that, for any term in ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}}$ , since b ≥ 2, then ${\displaystyle {\frac {b-1}{b^{k!}}}<{\frac {b-1}{b^{k}}}}$ , for all k (except for when n=1). Therefore, ${\displaystyle {\begin{aligned}\sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k!}}}<\sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k}}}\end{aligned}}}$  (since, even if n=1, all subsequent terms are smaller). In order to manipulate the indices so that k starts at 0, partial sum will be selected from within ${\displaystyle \sum _{k=n+1}^{\infty }{\frac {b-1}{b^{k}}}}$  (also less than the total value since it's a partial sum from a series whose terms are all positive). Choose the partial sum formed by starting at k = (n+1)! which follows from the motivation to write a new series with k=0, namely by noticing that ${\displaystyle b^{(n+1)!}=b^{(n+1)!}b^{0}}$ .
3. For the final inequality ${\displaystyle {\frac {b}{b^{(n+1)!}}}\leq {\frac {b^{n!}}{b^{(n+1)!}}}}$ , this particular inequality has been chosen (true because b ≥ 2, where equality follows if and only if n=1) because of the wish to manipulate ${\displaystyle {\frac {b}{b^{(n+1)!}}}}$  into something of the form ${\displaystyle {\frac {1}{b^{{\text{exponent}}\times n}}}}$ . This particular inequality allows the elimination of (n+1)! and the numerator, using the property that (n+1)! – n! = (n!)n, thus putting the denominator in ideal form for the substitution ${\displaystyle q_{n}=b^{n!}}$ .

Irrationality

Here the proof will show that the number ${\displaystyle ~x=c/d~,}$  where c and d are integers and ${\displaystyle ~d>0~,}$  cannot satisfy the inequalities that define a Liouville number. Since every rational number can be represented as such${\displaystyle ~c/d~,}$  the proof will show that no Liouville number can be rational.

More specifically, this proof shows that for any positive integer n large enough that ${\displaystyle ~2^{n-1}>d>0~}$  [equivalently, for any positive integer ${\displaystyle ~n>1+\log _{2}(d)~}$ )], no pair of integers ${\displaystyle ~(\,p,\,q\,)~}$  exists that simultaneously satisfies the pair of bracketing inequalities

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

If the claim is true, then the desired conclusion follows.

Let p and q be any integers with ${\displaystyle ~q>1~.}$  Then,

${\displaystyle \left|x-{\frac {\,p\,}{q}}\right|=\left|{\frac {\,c\,}{d}}-{\frac {\,p\,}{q}}\right|={\frac {\,|c\,q-d\,p|\,}{d\,q}}}$ 

If ${\displaystyle \left|c\,q-d\,p\right|=0~,}$  then

${\displaystyle \left|x-{\frac {\,p\,}{q}}\right|={\frac {\,|c\,q-d\,p|\,}{d\,q}}=0~,}$ 

meaning that such pair of integers ${\displaystyle ~(\,p,\,q\,)~}$  would violate the first inequality in the definition of a Liouville number, irrespective of any choice of n .

If, on the other hand, since ${\displaystyle ~\left|c\,q-d\,p\right|>0~,}$  then, since ${\displaystyle c\,q-d\,p}$  is an integer, we can assert the sharper inequality ${\displaystyle \left|c\,q-d\,p\right|\geq 1~.}$  From this it follows that

${\displaystyle \left|x-{\frac {\,p\,}{q}}\right|={\frac {\,|c\,q-d\,p|\,}{d\,q}}\geq {\frac {1}{\,d\,q\,}}}$ 

Now for any integer ${\displaystyle ~n>1+\log _{2}(d)~,}$  the last inequality above implies

${\displaystyle \left|x-{\frac {\,p\,}{q}}\right|\geq {\frac {1}{\,d\,q\,}}>{\frac {1}{\,2^{n-1}q\,}}\geq {\frac {1}{\;q^{n}\,}}~.}$ 

Therefore, in the case ${\displaystyle ~\left|c\,q-d\,p\right|>0~}$  such pair of integers ${\displaystyle ~(\,p,\,q\,)~}$  would violate the second inequality in the definition of a Liouville number, for some positive integer n.

Therefore, to conclude, there is no pair of integers ${\displaystyle ~(\,p,\,q\,)~,}$  with ${\displaystyle ~q>1~,}$  that would qualify such an ${\displaystyle ~x=c/d~,}$  as a Liouville number.

Hence a Liouville number, if it exists, cannot be rational.

(The section on Liouville's constant proves that Liouville numbers exist by exhibiting the construction of one. The proof given in this section implies that this number must be irrational.)

Uncountability

Consider, for example, the number

3.1400010000000000000000050000....

3.14(3 zeros)1(17 zeros)5(95 zeros)9(599 zeros)2(4319 zeros)6...

where the digits are zero except in positions n! where the digit equals the nth digit following the decimal point in the decimal expansion of π.

As shown in the section on the existence of Liouville numbers, this number, as well as any other non-terminating decimal with its non-zero digits similarly situated, satisfies the definition of a Liouville number. Since the set of all sequences of non-null digits has the cardinality of the continuum, the same thing occurs with the set of all Liouville numbers.

Moreover, the Liouville numbers form a dense subset of the set of real numbers.

Liouville numbers and measure

From the point of view of measure theory, the set of all Liouville numbers ${\displaystyle L}$  is small. More precisely, its Lebesgue measure, ${\displaystyle \lambda (L)}$ , is zero. The proof given follows some ideas by John C. Oxtoby.^[4]: 8

For positive integers ${\displaystyle n>2}$  and ${\displaystyle q\geq 2}$  set:

${\displaystyle V_{n,q}=\bigcup \limits _{p=-\infty }^{\infty }\left({\frac {p}{q}}-{\frac {1}{q^{n}}},{\frac {p}{q}}+{\frac {1}{q^{n}}}\right)}$ 

then

${\displaystyle L\subseteq \bigcup _{q=2}^{\infty }V_{n,q}.}$ 

Observe that for each positive integer ${\displaystyle n\geq 2}$  and ${\displaystyle m\geq 1}$ , then

${\displaystyle L\cap (-m,m)\subseteq \bigcup \limits _{q=2}^{\infty }V_{n,q}\cap (-m,m)\subseteq \bigcup \limits _{q=2}^{\infty }\bigcup \limits _{p=-mq}^{mq}\left({\frac {p}{q}}-{\frac {1}{q^{n}}},{\frac {p}{q}}+{\frac {1}{q^{n}}}\right).}$ 

Since

${\displaystyle \left|\left({\frac {p}{q}}+{\frac {1}{q^{n}}}\right)-\left({\frac {p}{q}}-{\frac {1}{q^{n}}}\right)\right|={\frac {2}{q^{n}}}}$ 

and ${\displaystyle n>2}$  then

${\displaystyle {\begin{aligned}\mu (L\cap (-m,\,m))&\leq \sum _{q=2}^{\infty }\sum _{p=-mq}^{mq}{\frac {2}{q^{n}}}=\sum _{q=2}^{\infty }{\frac {2(2mq+1)}{q^{n}}}\\[6pt]&\leq (4m+1)\sum _{q=2}^{\infty }{\frac {1}{q^{n-1}}}\leq (4m+1)\int _{1}^{\infty }{\frac {dq}{q^{n-1}}}\leq {\frac {4m+1}{n-2}}.\end{aligned}}}$ 

Now

${\displaystyle \lim _{n\to \infty }{\frac {4m+1}{n-2}}=0}$ 

and it follows that for each positive integer ${\displaystyle m}$ , ${\displaystyle L\cap (-m,m)}$  has Lebesgue measure zero. Consequently, so has ${\displaystyle L}$ .

In contrast, the Lebesgue measure of the set of all real transcendental numbers is infinite (since the set of algebraic numbers is a null set).

Structure of the set of Liouville numbers

For each positive integer n, set

${\displaystyle ~U_{n}=\bigcup \limits _{q=2}^{\infty }~\bigcup \limits _{p=-\infty }^{\infty }~\left\{x\in \mathbb {R} :0<\left|x-{\frac {p}{\,q\,}}\right|<{\frac {1}{\;q^{n}\,}}\right\}=\bigcup \limits _{q=2}^{\infty }~\bigcup \limits _{p=-\infty }^{\infty }~\left({\frac {p}{q}}-{\frac {1}{q^{n}}}~,~{\frac {p}{\,q\,}}+{\frac {1}{\;q^{n}\,}}\right)\setminus \left\{{\frac {p}{\,q\,}}\right\}~}$ 

The set of all Liouville numbers can thus be written as

${\displaystyle ~L~=~\bigcap \limits _{n=1}^{\infty }U_{n}~=~\bigcap \limits _{n\in \mathbb {N} _{1}}~\bigcup \limits _{q\geqslant 2}~\bigcup \limits _{p\in \mathbb {Z} }\,\left(\,\left(\,{\frac {\,p\,}{q}}-{\frac {1}{\;q^{n}\,}}~,~{\frac {\,p\,}{q}}+{\frac {1}{\;q^{n}\,}}\,\right)\setminus \left\{\,{\frac {\,p\,}{q}}\,\right\}\,\right)~.}$ 

Each ${\displaystyle ~U_{n}~}$  is an open set; as its closure contains all rationals (the ${\displaystyle ~p/q~}$  from each punctured interval), it is also a dense subset of real line. Since it is the intersection of countably many such open dense sets, L is comeagre, that is to say, it is a dense G_δ set.

Irrationality measure

The Liouville–Roth irrationality measure (irrationality exponent, approximation exponent, or Liouville–Roth constant) of a real number ${\displaystyle x}$  is a measure of how "closely" it can be approximated by rationals. Generalizing the definition of Liouville numbers, instead of allowing any ${\displaystyle n}$  in the power of ${\displaystyle q}$ , we find the largest possible value for ${\displaystyle \mu }$  such that ${\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}}$  is satisfied by an infinite number of coprime integer pairs ${\displaystyle (p,q)}$  with ${\displaystyle q>0}$ . This maximum value of ${\displaystyle \mu }$  is defined to be the irrationality measure of ${\displaystyle x}$ .^[5]: 246 For any value ${\displaystyle \mu }$  less than this upper bound, the infinite set of all rationals ${\displaystyle p/q}$  satisfying the above inequality yield an approximation of ${\displaystyle x}$ . Conversely, if ${\displaystyle \mu }$  is greater than the upper bound, then there are at most finitely many ${\displaystyle (p,q)}$  with ${\displaystyle q>0}$  that satisfy the inequality; thus, the opposite inequality holds for all larger values of ${\displaystyle q}$ . In other words, given the irrationality measure ${\displaystyle \mu }$  of a real number ${\displaystyle x}$ , whenever a rational approximation ${\displaystyle x\approx p/q}$ , ${\displaystyle p,q\in \mathbb {N} }$  yields ${\displaystyle n+1}$  exact decimal digits, then

${\displaystyle {\frac {1}{10^{n}}}\geq \left|x-{\frac {p}{q}}\right|\geq {\frac {1}{q^{\mu +\varepsilon }}}}$ 

for any ${\displaystyle \varepsilon >0}$ , except for at most a finite number of "lucky" pairs ${\displaystyle (p,q)}$ .

As a consequence of Dirichlet's approximation theorem every irrational number has irrationality measure at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers have an irrationality measure equal to 2.^[5]: 246

Below is a table of known upper and lower bounds for the irrationality measures of certain numbers.

Number ${\displaystyle x}$	Irrationality measure ${\displaystyle \mu (x)}$		Simple continued fraction ${\displaystyle [a_{0};a_{1},a_{2},...]}$	Notes
	Lower bound	Upper bound
Rational number ${\displaystyle {\frac {p}{q}}}$  where ${\displaystyle p,q\in \mathbb {Z} }$  and ${\displaystyle q\neq 0}$	1		Finite continued fraction.	Every rational number ${\displaystyle {\frac {p}{q}}}$  has an irrationality measure of exactly 1. Examples include 1, 2 and 0.5
Irrational algebraic number ${\displaystyle a}$	2		Infinite continued fraction. Periodic if quadratic irrational.	By the Thue–Siegel–Roth theorem the irrationality measure of any irrational algebraic number is exactly 2. Examples include square roots like ${\displaystyle {\sqrt {2}},{\sqrt {3}}}$  and ${\displaystyle {\sqrt {5}}}$  and the golden ratio ${\displaystyle \varphi }$ .
${\displaystyle e^{2/k},k\in \mathbb {Z} ^{+}}$	2		Infinite continued fraction.	If the elements ${\displaystyle a_{n}}$  of the continued fraction expansion of an irrational number ${\displaystyle x}$  satisfy ${\displaystyle a_{n}<cn+d}$  for positive ${\displaystyle c}$  and ${\displaystyle d}$ , the irrationality measure ${\displaystyle \mu (x)=2}$ . Examples include ${\displaystyle e}$  or ${\displaystyle I_{0}(1)/I_{1}(1)}$  where the continued fractions behave predictably: ${\displaystyle e=[2;1,2,1,1,4,1,1,6,1,1,...]}$  and ${\displaystyle I_{0}(1)/I_{1}(1)=[2;4,6,8,10,12,14,16,18,20,22...]}$
${\displaystyle \tanh \left({\frac {1}{k}}\right),k\in \mathbb {Z} ^{+}}$	2
${\displaystyle \tan \left({\frac {1}{k}}\right),k\in \mathbb {Z} ^{+}}$	2
${\displaystyle h_{q}(1)}$ ^[6]^[7]	2	2.49846...	Infinite continued fraction.	${\displaystyle q\in \{\pm 2,\pm 3,\pm 4,...\}}$ , ${\displaystyle h_{q}(1)}$  is a ${\displaystyle q}$ -harmonic series.
${\displaystyle {\text{ln}}_{q}(2)}$ ^[6]^[8]	2	2.93832...		${\displaystyle q\in \left\{\pm {\frac {1}{2}},\pm {\frac {1}{3}},\pm {\frac {1}{4}},...\right\}}$ , ${\displaystyle \ln _{q}(x)}$  is a ${\displaystyle q}$ -logarithm.
${\displaystyle \ln _{q}(1-z)}$ ^[6]^[8]	2	3.76338...		${\displaystyle q\in \left\{\pm {\frac {1}{2}},\pm {\frac {1}{3}},\pm {\frac {1}{4}},...\right\}}$ , ${\displaystyle 0<|z|\leq 1}$
${\displaystyle \ln(2)}$ ^[6]^[9]	2	3.57455...	${\displaystyle [0;1,2,3,1,6,3,1,1,2,1,...]}$
${\displaystyle \ln(3)}$ ^[6]^[10]	2	5.11620...	${\displaystyle [1;10,7,9,2,2,1,3,1,1,32,...]}$
${\displaystyle \zeta (3)}$ ^[6]	2	5.51389...	${\displaystyle [1;4,1,18,1,1,1,4,1,9,9,...]}$
${\displaystyle \pi ^{2}}$  and ${\displaystyle \zeta (2)}$ ^[6]^[11]	2	5.09541...	${\displaystyle [9;1,6,1,2,47,1,8,1,1,2,...]}$  and ${\displaystyle [1;1,1,1,4,2,4,7,1,4,2,...]}$	${\displaystyle \pi ^{2}}$  and ${\displaystyle \zeta (2)=\pi ^{2}/6}$  are linearly dependent over ${\displaystyle \mathbb {Q} }$ .
${\displaystyle \pi }$ ^[6]^[12]	2	7.10320...	${\displaystyle [3;7,15,1,292,1,1,1,2,1,3,...]}$	It has been proven that if the Flint Hills series ${\displaystyle \displaystyle \sum _{n=1}^{\infty }{\frac {\csc ^{2}n}{n^{3}}}}$  (where n is in radians) converges, then ${\displaystyle \pi }$ 's irrationality measure is at most 2.5;^[13]^[14] and that if it diverges, the irrationality measure is at least 2.5.^[15]
${\displaystyle \arctan(1/3)}$ ^[16]	2	6.09675...	${\displaystyle [0;3,9,3,1,5,1,6,3,1,2,...]}$	Of the form ${\displaystyle \arctan(1/k)}$
${\displaystyle \arctan(1/5)}$ ^[17]	2	4.788...	${\displaystyle [0;5,15,6,3,5,3,4,2,65,1,...]}$
${\displaystyle \arctan(1/6)}$ ^[17]	2	6.24...	${\displaystyle [0;6,18,7,1,1,4,5,62,2,1,...]}$
${\displaystyle \arctan(1/7)}$ ^[17]	2	4.076...	${\displaystyle [0;7,21,8,1,3,1,8,2,6,1,...]}$
${\displaystyle \arctan(1/10)}$ ^[17]	2	4.595...	${\displaystyle [0;10,30,12,1,1,7,3,2,1,3,...]}$
${\displaystyle \arctan(1/4)}$ ^[17]	2	5.793...	${\displaystyle [0;4,12,5,12,1,1,1,3,2,1,...]}$	Of the form ${\displaystyle \arctan(1/2^{k})}$
${\displaystyle \arctan(1/8)}$ ^[17]	2	3.673...	${\displaystyle [0;8,24,10,24,1,77,1,1,5,1,...]}$
${\displaystyle \arctan(1/16)}$ ^[17]	2	3.068...	${\displaystyle [0;16,48,20,49,1,4,1,3,1,1,...]}$
${\displaystyle \pi /{\sqrt {3}}}$ ^[18]^[19]	2	4.60105...	${\displaystyle [1;1,4,2,1,2,3,7,3,3,30,...]}$	Of the form ${\displaystyle {\sqrt {2k-1}}\arctan \left({\frac {\sqrt {2k-1}}{k-1}}\right)}$
${\displaystyle {\sqrt {7}}\arctan({{\sqrt {7}}/3})}$ ^[19]	2	3.94704...	${\displaystyle [1;1,10,2,1,1,2,3,6,1,3,...]}$
${\displaystyle {\sqrt {11}}\arctan({{\sqrt {11}}/5})}$ ^[19]	2	3.76069...	${\displaystyle [1;1,16,2,1,1,3,1,6,1,24,...]}$
${\displaystyle {\sqrt {15}}\arctan({{\sqrt {15}}/7})}$ ^[19]	2	3.66666...	${\displaystyle [1;1,22,2,1,1,5,2,3,10,2,...]}$
${\displaystyle {\sqrt {19}}\arctan({{\sqrt {19}}/9})}$ ^[19]	2	3.60809...	${\displaystyle [1;1,28,2,1,1,6,1,72,2,1,...]}$
${\displaystyle {\sqrt {23}}\arctan({{\sqrt {23}}/11})}$ ^[19]	2	3.56730...	${\displaystyle [1;1,34,2,1,1,8,1,1,5,2,...]}$
${\displaystyle {\sqrt {7}}\ln \left({\frac {4+{\sqrt {7}}}{3}}\right)}$ ^[19]	2	6.64610...	${\displaystyle [2;9,1,1,2,2,8,2,1,3,2,...]}$	Of the form ${\displaystyle {\sqrt {2k+1}}\ln \left({\frac {{\sqrt {2k+1}}+1}{{\sqrt {2k+1}}-1}}\right)}$
${\displaystyle {\sqrt {11}}\ln \left({\frac {6+{\sqrt {11}}}{5}}\right)}$ ^[19]	2	5.82337...	${\displaystyle [2;15,1,1,2,3,1,2,10,1,4,...]}$
${\displaystyle {\sqrt {13}}\ln \left({\frac {7+{\sqrt {13}}}{6}}\right)}$ ^[19]	2	3.51433...	${\displaystyle [2;18,1,1,2,4,2,5,33,6,2,...]}$
${\displaystyle {\sqrt {15}}\ln \left({\frac {8+{\sqrt {15}}}{7}}\right)}$ ^[19]	2	5.45248...	${\displaystyle [2;21,1,1,2,5,4,3,2,1,1,...]}$
${\displaystyle {\sqrt {17}}\ln \left({\frac {9+{\sqrt {17}}}{8}}\right)}$ ^[19]	2	3.47834...	${\displaystyle [2;24,1,1,2,6,92,3,3,1,16,...]}$
${\displaystyle {\sqrt {19}}\ln \left({\frac {10+{\sqrt {19}}}{9}}\right)}$ ^[19]	2	5.23162...	${\displaystyle [2;27,1,1,2,6,1,3,1,2,1,...]}$
${\displaystyle {\sqrt {21}}\ln \left({\frac {11+{\sqrt {21}}}{10}}\right)}$ ^[19]	2	3.45356...	${\displaystyle [2;30,1,1,2,7,1,1,3,4,63,...]}$
${\displaystyle {\sqrt {23}}\ln \left({\frac {12+{\sqrt {23}}}{11}}\right)}$ ^[19]	2	5.08120...	${\displaystyle [2;33,1,1,2,8,2,2,1,9,4,...]}$
${\displaystyle 5\ln(3/2)}$ ^[19]	2	3.43506...	${\displaystyle [2;36,1,1,2,9,8,5,1,38,1,...]}$
${\displaystyle {\frac {\pi }{\sqrt {3}}}\pm \ln(3)}$ ^[17]	2	4.5586...	${\displaystyle [2;1,10,2,2,1,1,17,1,4,1,...]}$  and ${\displaystyle [0;1,2,1,1,22,14,3,1,1,1,...]}$
${\displaystyle {\sqrt {3}}\ln(2+{\sqrt {3}})\pm {\frac {\pi }{\sqrt {3}}}}$ ^[17]	2	6.1382...	${\displaystyle [4;10,1,1,5,7,2,2,1,31,2,...]}$  and ${\displaystyle [0;2,7,7,1,1,1,3,9,9,1,...]}$
${\displaystyle \ln(5)+{\frac {\pi }{2}}}$ ^[17]	2	59.976...	${\displaystyle [3;5,1,1,4,1,2,19,1,3,...]}$
${\displaystyle T_{2}(1/b),b\geq 2}$ ^[20]	2	4	Infinite continued fraction.	${\displaystyle T_{2}(1/b):=\sum _{n=1}^{\infty }t_{n}b^{n-1}}$  where ${\displaystyle t_{n}}$  is the ${\displaystyle n}$ -th term of the Thue–Morse sequence.
Champernowne constants ${\displaystyle C_{b}}$  in base ${\displaystyle b\geq 2}$ ^[21]	${\displaystyle b}$		Infinite continued fraction.	Examples include ${\displaystyle C_{10}=0.1234567891011...=[0;8,9,1,149083,1,...]}$
Liouville numbers ${\displaystyle L}$	${\displaystyle \infty }$		Infinite continued fraction, not behaving predictable.	The Liouville numbers are precisely those numbers having infinite irrationality measure.^[5]^: 248

Irrationality base

The irrationality base is a measure of irrationality introduced by J. Sondow^[22] as an irrationality measure for Liouville numbers. It is defined as follows:

Let ${\displaystyle \alpha }$  be an irrational number. If there exists a real number ${\displaystyle \beta \geq 1}$  with the property that for any ${\displaystyle \varepsilon >0}$ , there is a positive integer ${\displaystyle q(\varepsilon )}$  such that

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|>{\frac {1}{(\beta +\varepsilon )^{q}}}{\text{ for all integers }}p,q{\text{ with }}q\geq q(\varepsilon )}$ ,

then ${\displaystyle \beta }$  is called the irrationality base of ${\displaystyle \alpha }$  and is represented as ${\displaystyle \beta (\alpha )}$ 

If no such ${\displaystyle \beta }$  exists, then ${\displaystyle \alpha }$  is called a super Liouville number.

Example: The series ${\displaystyle \varepsilon _{2e}=1+{\frac {1}{2^{1}}}+{\frac {1}{4^{2^{1}}}}+{\frac {1}{8^{4^{2^{1}}}}}+{\frac {1}{16^{8^{4^{2^{1}}}}}}+{\frac {1}{32^{16^{8^{4^{2^{1}}}}}}}+\ldots }$  is a super Liouville number, while the series ${\displaystyle \tau _{2}=\sum _{n=1}^{\infty }{\frac {1}{^{n}2}}={\frac {1}{2}}+{\frac {1}{2^{2}}}+{\frac {1}{2^{2^{2}}}}+{\frac {1}{2^{2^{2^{2}}}}}+{\frac {1}{2^{2^{2^{2^{2}}}}}}+\ldots }$  is a Liouville number with irrationality base 2. (${\displaystyle {^{b}a}}$  represents tetration.)

Liouville numbers and transcendence

Establishing that a given number is a Liouville number provides a useful tool for proving a given number is transcendental. However, not every transcendental number is a Liouville number. The terms in the continued fraction expansion of every Liouville number are unbounded; using a counting argument, one can then show that there must be uncountably many transcendental numbers which are not Liouville. Using the explicit continued fraction expansion of e, one can show that e is an example of a transcendental number that is not Liouville. Mahler proved in 1953 that π is another such example.^[23]

The proof proceeds by first establishing a property of irrational algebraic numbers. This property essentially says that irrational algebraic numbers cannot be well approximated by rational numbers, where the condition for "well approximated" becomes more stringent for larger denominators. A Liouville number is irrational but does not have this property, so it can't be algebraic and must be transcendental. The following lemma is usually known as Liouville's theorem (on diophantine approximation), there being several results known as Liouville's theorem.

Below, the proof will show that no Liouville number can be algebraic.

Lemma: If α is an irrational number which is the root of an irreducible polynomial f of degree n > 0 with integer coefficients, then there exists a real number A > 0 such that, for all integers p, q, with q > 0,

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|>{\frac {A}{q^{n}}}}$ 

Proof of Lemma: Let M be the maximum value of |f ′(x)| (the absolute value of the derivative of f) over the interval [α − 1, α + 1]. Let α₁, α₂, ..., α_m be the distinct roots of f which differ from α. Select some value A > 0 satisfying

${\displaystyle A<\min \left(1,{\frac {1}{M}},\left|\alpha -\alpha _{1}\right|,\left|\alpha -\alpha _{2}\right|,\ldots ,\left|\alpha -\alpha _{m}\right|\right)}$ 

Now assume that there exist some integers p, q contradicting the lemma. Then

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|\leq {\frac {A}{q^{n}}}\leq A<\min \left(1,{\frac {1}{M}},\left|\alpha -\alpha _{1}\right|,\left|\alpha -\alpha _{2}\right|,\ldots ,\left|\alpha -\alpha _{m}\right|\right)}$ 

Then p/q is in the interval [α − 1, α + 1]; and p/q is not in {α₁, α₂, ..., α_m}, so p/q is not a root of f; and there is no root of f between α and p/q.

By the mean value theorem, there exists an x₀ between p/q and α such that

${\displaystyle f(\alpha )-f({\tfrac {p}{q}})=(\alpha -{\frac {p}{q}})\cdot f'(x_{0})}$ 

Since α is a root of f but p/q is not, it follows that |f ′(x₀)| > 0 and can rearrange:

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|={\frac {\left|f(\alpha )-f({\tfrac {p}{q}})\right|}{|f'(x_{0})|}}=\left|{\frac {f({\tfrac {p}{q}})}{f'(x_{0})}}\right|}$ 

Now, f is of the form ${\displaystyle \sum _{i=0}^{n}}$  c_i xⁱ where each c_i is an integer; so expressing |f(p/q)| as

${\displaystyle \left|f\left({\frac {p}{q}}\right)\right|=\left|\sum _{i=0}^{n}c_{i}p^{i}q^{-i}\right|={\frac {1}{q^{n}}}\left|\sum _{i=0}^{n}c_{i}p^{i}q^{n-i}\right|\geq {\frac {1}{q^{n}}}}$ 

N.B. the last inequality holds because p/q is not a root of f and the c_i are integers.

Thus, |f(p/q)| ≥ 1/qⁿ. Since |f ′(x₀)| ≤ M by the definition of M, and 1/M > A by the definition of A, then

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|=\left|{\frac {f({\tfrac {p}{q}})}{f'(x_{0})}}\right|\geq {\frac {1}{Mq^{n}}}>{\frac {A}{q^{n}}}\geq \left|\alpha -{\frac {p}{q}}\right|}$ 

which is a contradiction; therefore, no such p, q exist; proving the lemma.

Proof of assertion: As a consequence of this lemma, let x be a Liouville number; as noted in the article text, x is then irrational. If x is algebraic, then by the lemma, there exists some integer n and some positive real A such that for all p, q

${\displaystyle \left|x-{\frac {p}{q}}\right|>{\frac {A}{q^{n}}}}$ 

Let r be a positive integer such that 1/(2^r) ≤ A. If m = r + n, and since x is a Liouville number, then there exist integers a, b where b > 1 such that

${\displaystyle \left|x-{\frac {a}{b}}\right|<{\frac {1}{b^{m}}}={\frac {1}{b^{r+n}}}={\frac {1}{b^{r}b^{n}}}\leq {\frac {1}{2^{r}}}{\frac {1}{b^{n}}}\leq {\frac {A}{b^{n}}}}$ 

which contradicts the lemma. Hence a Liouville number cannot be algebraic, and therefore must be transcendental.

See also

• Brjuno number
• Diophantine approximation

References

1. Joseph Liouville (May 1844). "Mémoires et communications". Comptes rendus de l'Académie des Sciences (in French). 18 (20, 21): 883–885, 910–911.
2. Baker, Alan (1990). Transcendental Number Theory (paperback ed.). Cambridge University Press. p. 1.
3. Baker 1990, p. 86.
4. Oxtoby, John C. (1980). Measure and Category. Graduate Texts in Mathematics. Vol. 2 (Second ed.). New York-Berlin: Springer-Verlag. doi:10.1007/978-1-4684-9339-9. ISBN 0-387-90508-1. MR 0584443.
5. Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193. Cambridge: Cambridge University Press. doi:10.1017/CBO9781139017732. ISBN 978-0-521-11169-0. MR 2953186. Zbl 1260.11001.
6. Weisstein, Eric W. "Irrationality Measure". mathworld.wolfram.com. Retrieved 2020-10-14.
7. Zudilin, Wadim (2002-04-01). "Remarks on irrationality of q-harmonic series". Manuscripta Mathematica. 107 (4): 463–477. doi:10.1007/s002290200249. ISSN 1432-1785. S2CID 120782644.
8. Matala-aho, Tapani; Väänänen, Keijo; Zudilin, Wadim (2006). "New irrationality measures for 𝑞-logarithms". Mathematics of Computation. 75 (254): 879–889. doi:10.1090/S0025-5718-05-01812-0. ISSN 0025-5718.
9. Nesterenko, Yu. V. (2010-10-01). "On the irrationality exponent of the number ln 2". Mathematical Notes. 88 (3): 530–543. doi:10.1134/S0001434610090257. ISSN 1573-8876. S2CID 120685006.
10. "Symmetrized polynomials in a problem of estimating of the irrationality measure of number ln 3". www.mathnet.ru. Retrieved 2020-10-14.
11. Zudilin, Wadim (2014-06-01). "Two hypergeometric tales and a new irrationality measure of ζ(2)". Annales mathématiques du Québec. 38 (1): 101–117. arXiv:1310.1526. doi:10.1007/s40316-014-0016-0. ISSN 2195-4763. S2CID 119154009.
12. Zeilberger, Doron; Zudilin, Wadim (2020-01-07). "The irrationality measure of π is at most 7.103205334137…". Moscow Journal of Combinatorics and Number Theory. 9 (4): 407–419. arXiv:1912.06345. doi:10.2140/moscow.2020.9.407. S2CID 209370638.
13. Alekseyev, Max A. (2011). "On convergence of the Flint Hills series". arXiv:1104.5100 [math.CA].
14. Weisstein, Eric W. "Flint Hills Series". MathWorld.
15. Meiburg, Alex (2022). "Bounds on Irrationality Measures and the Flint-Hills Series". arXiv:2208.13356 [math.NT].
16. Salikhov, V. Kh.; Bashmakova, M. G. (2019-01-01). "On Irrationality Measure of arctan 1/3". Russian Mathematics. 63 (1): 61–66. doi:10.3103/S1066369X19010079. ISSN 1934-810X. S2CID 195131482.
17. Tomashevskaya, E. B. "On the irrationality measure of the number log 5+pi/2 and some other numbers". www.mathnet.ru. Retrieved 2020-10-14.
18. Androsenko, V. A. (2015). "Irrationality measure of the number \frac{\pi}{\sqrt{3}}". Izvestiya: Mathematics. 79 (1): 1–17. doi:10.1070/im2015v079n01abeh002731. ISSN 1064-5632. S2CID 123775303.
19. Polyanskii, A. A. (2018-03-01). "On the Irrationality Measures of Certain Numbers. II". Mathematical Notes. 103 (3): 626–634. doi:10.1134/S0001434618030306. ISSN 1573-8876. S2CID 125251520.
20. Adamczewski, Boris; Rivoal, Tanguy (2009). "Irrationality measures for some automatic real numbers". Mathematical Proceedings of the Cambridge Philosophical Society. 147 (3): 659–678. Bibcode:2009MPCPS.147..659A. doi:10.1017/S0305004109002643. ISSN 1469-8064. S2CID 1689323.
21. Amou, Masaaki (1991-02-01). "Approximation to certain transcendental decimal fractions by algebraic numbers". Journal of Number Theory. 37 (2): 231–241. doi:10.1016/S0022-314X(05)80039-3. ISSN 0022-314X.
22. Sondow, Jonathan (2004). "Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik". arXiv:math/0406300.
23. The irrationality measure of π does not exceed 7.6304, according to Weisstein, Eric W. "Irrationality Measure". MathWorld.

External links

• The Beginning of Transcendental Numbers