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{{short description|Class of irrational numbers}}
In [[Number theory|number theory]], a '''Liouville number''' is a [[Real number|real number]] <math>x</math> with the property that, for every positive [[Integer|integer]] <math>n</math>, there exists a pair of integers <math>(p,q)</math> with <math>q>1</math> such that
<math display=block>0 < \left|x-\frac{p}{q}\right| < \frac{1}{q^{n}} .</math>

Liouville numbers are "almost [[Rational number|rational]]", and can thus be approximated "quite closely" by [[Sequence|sequence]]s of rational numbers. Precisely, these are [[Transcendental number|transcendental number]]s that can be more closely approximated by rational numbers than any [[Algebraic number|algebraic]] [[Irrational number|irrational number]] can be. In 1844, [[Biography:Joseph Liouville|Joseph Liouville]] showed that all Liouville numbers are transcendental,<ref>{{cite journal <!--DUPLICATE| url=https://gallica.bnf.fr/ark:/12148/cb343481087/date1844.liste--> | url=http://www.bibnum.education.fr/mathematiques/theorie-des-nombres/propos-de-l-existence-des-nombres-transcendants | author=Joseph Liouville | title=Mémoires et communications | journal=Comptes rendus de l'Académie des Sciences | volume=18 | number=20,21 | pages=883&ndash;885,910&ndash;911 | date=May 1844 | language=French}}</ref> thus establishing the existence of transcendental numbers for the first time.<ref>
{{cite book
 |first=Alan |last=Baker
 |year=1990
 |title=Transcendental Number Theory
 |edition=paperback
 |publisher=Cambridge University Press
 |page=1
}}
</ref>
It is known that [[Pi|{{pi}}]] and {{mvar|e}} are not Liouville numbers.{{sfn|Baker|1990|p=86}}

== The existence of Liouville numbers (Liouville's constant) ==
Liouville numbers can be shown to exist by an explicit construction.

For any integer <math>b\ge 2</math> and any sequence of integers <math>(a_1,a_2,\dots)</math> such that <math>a_k\in\{0,1,2,\dots,b-1\}</math> for all <math>k</math> and <math>a_k\ne 0</math> for infinitely many <math>k</math>, define the number
<math display=block>x = \sum_{k=1}^\infty \frac{a_k}{b^{k!}}.</math>

In the special case when <math>b=10</math>, and <math>a_k=1</math> for all <math>k</math>, the resulting number <math>x</math> is called '''Liouville's constant:'''

:''L'' = 0.'''11'''000'''1'''00000000000000000'''1'''0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000'''1'''...

It follows from the definition of <math>x</math> that its base-<math>b</math> representation is

:<math>x = \left(0.a_{1}a_{2}000a_{3}00000000000000000a_{4}0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a_{5}\ldots\right)_b\;</math>

where the <math>n</math>th term is in the <math>(n!)</math>th place.

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

Now, for any integer <math>n\ge 1</math>, <math>p_n</math> and <math>q_n</math> can be defined as follows:
<math display=block>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!}} \; .</math>

Then,
<math display="block">\begin{align}
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]
& \le \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)!}} \le \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{align}</math>

Therefore, any such <math>x</math> is a Liouville number.

=== Notes on the proof ===
# The inequality <math>\sum_{k=n+1}^\infty \frac{a_k}{b^{k!}} \le \sum_{k=n+1}^\infty \frac{b-1}{b^{k!}}</math> follows since ''a''<sub>''k''</sub>&nbsp;∈&nbsp;{0,&nbsp;1,&nbsp;2,&nbsp;..., ''b''−1} for all ''k'', so at most ''a''<sub>''k''</sub>&nbsp;= ''b''−1. The largest possible sum would occur if the sequence of integers (''a''<sub>1</sub>, ''a''<sub>2</sub>,&nbsp;...) were (''b''−1, ''b''−1, ...), i.e. ''a''<sub>''k''</sub>&nbsp;= ''b''−1, for all ''k''. <math>\sum_{k=n+1}^\infty \frac{a_k}{b^{k!}}</math> will thus be less than or equal to this largest possible sum.
# The strong inequality <math>\begin{align}
\sum_{k=n+1}^\infty \frac{b-1}{b^{k!}} < \sum_{k=(n+1)!}^\infty \frac{b-1}{b^k}
\end{align}</math> follows from the motivation to eliminate the [[Series (mathematics)|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 <math>\sum_{k=0}^\infty \frac{1}{b^{k}} = \frac{b}{b-1}</math> (the [[Geometric series|geometric series]] formula); therefore, if an inequality can be found from <math>\sum_{k=n+1}^\infty \frac{a_k}{b^{k!}}</math> that introduces a series with (''b''−1) in the numerator, and if the denominator term can be further reduced from <math>b^{k!}</math>to <math>b^{k}</math>, as well as shifting the series indices from 0 to <math>\infty</math>, then both series and (''b''−1) terms will be eliminated, getting closer to a fraction of the form <math>\frac{1}{b^{\text{exponent}\times n}}</math>, which is the end-goal of the proof. This motivation is increased here by selecting now from the sum <math>\sum_{k=n+1}^\infty \frac{b-1}{b^{k!}}</math> a partial sum. Observe that, for any term in <math>\sum_{k=n+1}^\infty \frac{b-1}{b^{k!}}</math>, since ''b'' ≥ 2, then <math>\frac{b-1}{b^{k!}} < \frac{b-1}{b^{k}}</math>, for all ''k'' (except for when ''n''=1). Therefore, <math>\begin{align}
\sum_{k=n+1}^\infty \frac{b-1}{b^{k!}} < \sum_{k=n+1}^\infty \frac{b-1}{b^k}
\end{align}</math> (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 <math>
\sum_{k=n+1}^\infty \frac{b-1}{b^k}
</math> (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 <math>b^{(n+1)!} = b^{(n+1)!}b^0</math>.
#For the final inequality <math>\frac{b}{b^{(n+1)!}} \le \frac{b^{n!}}{b^{(n+1)!}}</math>, this particular inequality has been chosen (true because ''b'' ≥ 2, where equality follows [[If and only if|if and only if]] ''n''=1) because of the wish to manipulate <math>\frac{b}{b^{(n+1)!}}</math> into something of the form <math>\frac{1}{b^{\text{exponent}\times n}}</math>. 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 <math>q_n = b^{n!}</math>.

== Irrationality ==
Here the proof will show that the number <math>~ x = c / d ~,</math> where {{mvar|c}} and {{mvar|d}} are integers and <math>~ d > 0 ~,</math> cannot satisfy the inequalities that define a Liouville number. Since every [[Rational number|rational number]] can be represented as such<math>~ c / d ~,</math> the proof will show that '''no Liouville number can be rational'''.

More specifically, this proof shows that for any positive integer {{mvar|n}} large enough that <math>~ 2^{n - 1} > d > 0~</math> [equivalently, for any positive integer <math>~ n > 1 + \log_2(d) ~</math>)], no pair of integers <math>~(\,p,\,q\,)~</math> exists that simultaneously satisfies the pair of bracketing inequalities

:<math>0 < \left|x - \frac{\,p\,}{q}\right| < \frac{1}{\;q^n\,}~.</math>

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

Let {{mvar|p}} and {{mvar|q}} be any integers with <math>~q > 1~.</math> Then,

:<math> \left| x - \frac{\,p\,}{q} \right| = \left| \frac{\,c\,}{d} - \frac{\,p\,}{q}  \right| = \frac{\,|c\,q - d\,p|\,}{ d\,q }</math>

If <math>  \left| c\,q - d\,p \right| = 0~,</math> then

:<math>\left| x - \frac{\,p\,}{q}\right|= \frac{\,|c\,q - d\,p|\,}{ d\,q } = 0 ~,</math>

meaning that such pair of integers <math>~(\,p,\,q\,)~</math> would violate the ''first'' inequality in the definition of a Liouville number, irrespective of any choice of {{mvar|n}}&nbsp;.

If, on the other hand, since <math>~\left| c\,q - d\,p \right| > 0 ~,</math> then, since <math>c\,q - d\,p</math>  is an integer, we can assert the sharper inequality <math>\left| c\,q - d\,p \right| \ge 1 ~.</math>  From this it follows that

:<math>\left| x - \frac{\,p\,}{q}\right|= \frac{\,| c\,q - d\,p |\,}{d\,q} \ge \frac{1}{\,d\,q\,}</math>

Now for any integer <math>~n > 1 + \log_2(d)~,</math> the last inequality above implies

:<math>\left| x - \frac{\,p\,}{q} \right| \ge \frac{1}{\,d\,q\,} > \frac{1}{\,2^{n-1}q\,} \ge \frac{1}{\;q^n\,} ~.</math>

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

Therefore, to conclude, there is no pair of integers <math>~(\,p,\,q\,)~,</math> with <math>~ q > 1 ~,</math> that would qualify such an <math>~ x = c / d ~,</math> 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 number|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 ''n''th digit following the decimal point in the decimal expansion of&nbsp;{{pi}}.

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|cardinality of the continuum]], the same thing occurs with the set of all Liouville numbers.

Moreover, the Liouville numbers form a [[Dense set|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 <math>L</math> is small. More precisely, its [[Lebesgue measure]], <math>\lambda(L)</math>, is zero. The proof given follows some ideas by [[Biography:John C. Oxtoby|John C. Oxtoby]].<ref name="oxtoby">{{Cite book | last = Oxtoby | first = John C. | year = 1980 | title = Measure and Category | series = Graduate Texts in Mathematics | volume = 2 | edition = Second | publisher = Springer-Verlag | isbn = 0-387-90508-1 | location = New York-Berlin | mr=0584443 | doi=10.1007/978-1-4684-9339-9}}</ref>{{Rp|8}}

For positive integers <math>n>2</math> and <math>q\geq2</math> set:

:<math>V_{n,q}=\bigcup\limits_{p=-\infty}^\infty \left(\frac{p}{q}-\frac{1}{q^n},\frac{p}{q}+\frac{1}{q^n}\right)</math>

then

:<math>L\subseteq \bigcup_{q=2}^\infty V_{n,q}.</math>

Observe that for each positive integer <math>n\geq2</math> and <math>m\geq1</math>, then

:<math>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).</math>

Since

: <math> \left|\left(\frac{p}{q}+\frac{1}{q^n}\right)-\left(\frac{p}{q}-\frac{1}{q^n}\right)\right|=\frac{2}{q^n}</math>

and <math>n>2</math> then

: <math>
\begin{align}
\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^\infty_1 \frac{dq}{q^{n-1}}\leq\frac{4m+1}{n-2}.
\end{align}
</math>

Now

:<math>\lim_{n\to\infty}\frac{4m+1}{n-2}=0</math>

and it follows that for each positive integer <math>m</math>, <math>L\cap (-m,m)</math> has Lebesgue measure zero. Consequently, so has <math>L</math>.

In contrast, the Lebesgue measure of the set of ''all'' real transcendental numbers is [[Infinity|infinite]] (since the set of algebraic numbers is a [[Null set|null set]]).

One could show even more - the set of Liouville numbers has [[Hausdorff dimension]] 0 (a property strictly stronger than having Lebesgue measure 0).

==Structure of the set of Liouville numbers==
For each positive integer {{mvar|n}}, set

:<math>~ 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\} ~</math>
The set of all Liouville numbers can thus be written as

:<math>~ 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) ~.</math>

Each <math>~ U_n ~</math> is an [[Open set|open set]]; as its closure contains all rationals (the <math>~p / q~</math> from each punctured interval), it is also a [[Dense set|dense]] subset of real line. Since it is the intersection of countably many such open dense sets, {{mvar|L}} is [[Meagre set|comeagre]], that is to say, it is a ''dense'' G<sub>δ</sub> set.

== Irrationality measure ==
The '''Liouville–Roth irrationality measure''' ('''irrationality exponent,''' '''approximation exponent,''' or '''Liouville–Roth constant''') of a real number <math>x</math> is a measure of how "closely" it can be approximated by rationals. Generalizing the definition of Liouville numbers, instead of allowing any <math>n</math> in the power of <math>q</math>, we find the largest possible value for <math>\mu</math> such that <math>0< \left| x- \frac{p}{q} \right| < \frac{1}{q^\mu} </math> is satisfied by an infinite number of coprime integer pairs <math>(p,q)</math> with <math>q>0</math>. This maximum value of <math>\mu</math> is defined to be the irrationality measure of <math>x</math>.<ref name=bugeaud>{{cite book | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher={{wipe|Cambridge University Press}} | year=2012 | isbn=978-0-521-11169-0 | zbl=1260.11001 | mr=2953186 | doi=10.1017/CBO9781139017732}}</ref>{{rp|246}}  For any value <math>\mu</math> less than this upper bound, the infinite set of all rationals <math>p/q</math> satisfying the above inequality yield an approximation of <math>x</math>. Conversely, if <math>\mu</math> is greater than the upper bound, then there are at most finitely many <math>(p,q)</math> with <math>q>0</math> that satisfy the inequality; thus, the opposite inequality holds for all larger values of <math>q</math>. In other words, given the irrationality measure <math>\mu</math> of a real number <math>x</math>, whenever a rational approximation <math>x\approx p/q</math>, <math>p,q\in\N</math> yields <math>n+1</math> exact decimal digits, then

:<math>\frac{1}{10^n} \ge \left| x- \frac{p}{q} \right| \ge \frac{1}{q^{\mu+\varepsilon}} </math>

for any <math>\varepsilon >0</math>, except for at most a finite number of "lucky"  pairs <math>(p,q)</math>.

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.<ref name="bugeaud" />{{rp|246}}

Below is a table of known upper and lower bounds for the irrationality measures of certain numbers.
{| class="wikitable sortable"
|+
! rowspan="2" |Number <math>x</math>
! colspan="2" |Irrationality measure <math>\mu(x)</math>
! rowspan="2" |Simple continued fraction <math>[a_0;a_1,a_2,...]</math>
! rowspan="2" |Notes
|-
!Lower bound
!Upper bound
|-
|[[Rational number]] <math>\frac{p}q</math> where <math>p,q \in \mathbb{Z}</math> and <math>q\neq0</math>
| colspan="2" style="text-align: center;" |1
|Finite [[Continued fraction|continued fraction]].
|Every rational number <math>\frac{p}q</math> has an irrationality measure of exactly 1.

Examples include 1, 2 and 0.5 
|-
|Irrational [[Algebraic number|algebraic number]] <math>a</math>
| colspan="2" style="text-align: center;" |2
|Infinite continued fraction. Periodic if [[Quadratic irrational number|quadratic irrational]].
|By the Thue–Siegel–Roth theorem the irrationality measure of any irrational algebraic number is exactly 2. Examples include [[Square root|square root]]s like <math>\sqrt{2}, \sqrt{3}</math> and <math>\sqrt{5}</math> and the [[Golden ratio|golden ratio]] <math>\varphi</math>.
|-
|<math>e^{2/k}, k\in\mathbb{Z}^+</math>
| colspan="2" style="text-align: center;" |2
| rowspan="3" |Infinite continued fraction.
| rowspan="3" |If the elements <math>a_n</math> of the continued fraction expansion of an irrational number <math>x</math> satisfy <math>a_n<cn+d</math> for positive <math>c</math> and <math>d</math>, the irrationality measure <math>\mu(x)=2</math>.

Examples include <math>e</math> or <math>I_0(1)/I_1(1)</math> where the continued fractions behave predictably:

<math>e=[2;1,2,1,1,4,1,1,6,1,1,...]</math>  and <math>I_0(1)/I_1(1)=[2;4,6,8,10,12,14,16,18,20,22...]</math>
|-
|<math>\tanh\left(\frac{1}{k}\right), k\in\mathbb{Z}^+</math>
| colspan="2" style="text-align: center;" |2
|-
|<math>\tan\left(\frac{1}{k}\right), k\in\mathbb{Z}^+</math>
| colspan="2" style="text-align: center;" |2
|-
|<math>h_q(1)</math><ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Irrationality Measure|url=https://mathworld.wolfram.com/IrrationalityMeasure.html|access-date=2020-10-14|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite journal|last=Zudilin|first=Wadim|date=2002-04-01|title=Remarks on irrationality of q-harmonic series|url=https://doi.org/10.1007/s002290200249|journal=Manuscripta Mathematica|language=en|volume=107|issue=4|pages=463–477|doi=10.1007/s002290200249|s2cid=120782644|issn=1432-1785}}</ref>
|2
|2.49846...
| rowspan="3" |Infinite continued fraction.
|<math>q\in\{\pm2,\pm3,\pm4,...\}</math>, <math>h_q(1)</math> is a <math>q</math>-harmonic series.
|-
|<math>\text{ln}_q(2)</math><ref name=":0" /><ref name=":1">{{Cite journal|last1=Matala-aho|first1=Tapani|last2=Väänänen|first2=Keijo|last3=Zudilin|first3=Wadim|date=2006|title=New irrationality measures for 𝑞-logarithms|url=https://www.ams.org/mcom/2006-75-254/S0025-5718-05-01812-0/|journal=Mathematics of Computation|language=en|volume=75|issue=254|pages=879–889|doi=10.1090/S0025-5718-05-01812-0|issn=0025-5718|doi-access=free|hdl=1959.13/934868|hdl-access=free}}</ref>
|2
|2.93832...
|<math>q\in\left\{\pm\frac{1}{2},\pm\frac{1}{3},\pm\frac{1}{4},...\right\}</math>, <math>\ln_q(x)</math> is a <math>q</math>-logarithm.
|-
|<math>\ln_q(1-z)</math><ref name=":0" /><ref name=":1" />
|2
|3.76338...
|<math>q\in\left\{\pm\frac{1}{2},\pm\frac{1}{3},\pm\frac{1}{4},...\right\}</math>, <math>0<|z|\leq1</math>
|-
|<math>\ln(2)</math><ref name=":0" /><ref>{{Cite journal|last=Nesterenko|first=Yu. V.|date=2010-10-01|title=On the irrationality exponent of the number ln 2|url=https://doi.org/10.1134/S0001434610090257|journal=Mathematical Notes|language=en|volume=88|issue=3|pages=530–543|doi=10.1134/S0001434610090257|s2cid=120685006|issn=1573-8876}}</ref>
|2
|3.57455...
|<math>[0;1,2,3,1,6,3,1,1,2,1,...]</math>
|
|-
|<math>\ln(3)</math><ref name=":0" /><ref>{{Cite web|title=Symmetrized polynomials in a problem of estimating of the irrationality measure of number ln 3|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=cheb&paperid=619&option_lang=eng|access-date=2020-10-14|website=www.mathnet.ru}}</ref>
|2
|5.11620...
|<math>[1;10,7,9,2,2,1,3,1,1,32,...]</math>
|
|-
|<math>\zeta(3)</math><ref name=":0" />
|2
|5.51389...
|<math>[1;4,1,18,1,1,1,4,1,9,9,...]</math>
|
|-
|<math>\pi^2</math> and <math>\zeta(2)</math><ref name=":0" /><ref>{{Cite journal|last=Zudilin|first=Wadim|date=2014-06-01|title=Two hypergeometric tales and a new irrationality measure of ''ζ''(2)|journal=Annales mathématiques du Québec|volume=38|issue=1|pages=101–117|doi=10.1007/s40316-014-0016-0|arxiv=1310.1526|s2cid=119154009|issn=2195-4763}}</ref>
|2
|5.09541...
|<math>[9;1,6,1,2,47,1,8,1,1,2,...]</math> and

<math>[1;1,1,1,4,2,4,7,1,4,2,...]</math>
|<math>\pi^2</math> and <math>\zeta(2)=\pi^2/6</math> are linearly dependent over <math>\mathbb{Q}</math>.
|-
|<math>\pi</math><ref name=":0" /><ref>{{cite journal|last1=Zeilberger|first1=Doron|last2=Zudilin|first2=Wadim|date=2020-01-07|title=The irrationality measure of ''π'' is at most 7.103205334137...|journal=Moscow Journal of Combinatorics and Number Theory|volume=9|issue=4|pages=407–419|doi=10.2140/moscow.2020.9.407|arxiv=1912.06345|s2cid=209370638}}</ref>
|2
|7.10320...
|<math>[3;7,15,1,292,1,1,1,2,1,3,...]</math>
|It has been proven that if the Flint Hills series <math>\displaystyle\sum^\infty_{n=1}\frac{\csc^2 n}{n^3}</math> (where ''n'' is in radians) converges, then <math>\pi</math>'s irrationality measure is at most 2.5;<ref>{{cite arXiv |first=Max A. |last=Alekseyev |title=On convergence of the Flint Hills series |eprint=1104.5100 |date=2011 |class=math.CA }}</ref><ref>{{MathWorld|FlintHillsSeries|Flint Hills Series}}</ref> and that if it diverges, the irrationality measure is at least 2.5.<ref>{{cite arXiv |first=Alex |last=Meiburg| title=Bounds on Irrationality Measures and the Flint-Hills Series|eprint=2208.13356| date=2022 |class=math.NT}}</ref>
|-
|<math>\arctan(1/3)</math><ref>{{Cite journal|last1=Salikhov|first1=V. Kh.|last2=Bashmakova|first2=M. G.|date=2019-01-01|title=On Irrationality Measure of arctan 1/3|url=https://doi.org/10.3103/S1066369X19010079|journal=Russian Mathematics|language=en|volume=63|issue=1|pages=61–66|doi=10.3103/S1066369X19010079|s2cid=195131482|issn=1934-810X}}</ref>
|2
|6.09675...
|<math>[0;3,9,3,1,5,1,6,3,1,2,...]</math>
| rowspan="5" |Of the form <math>\arctan(1/k)</math>
|-
|<math>\arctan(1/5)</math><ref name=":2">{{Cite web|last=Tomashevskaya|first=E. B.|title=On the irrationality measure of the number log 5+pi/2 and some other numbers|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=cheb&paperid=245&option_lang=eng|access-date=2020-10-14|website=www.mathnet.ru}}</ref>
|2
|4.788...
|<math>[0;5,15,6,3,5,3,4,2,65,1,...]</math>
|-
|<math>\arctan(1/6)</math><ref name=":2" />
|2
|6.24...
|<math>[0;6,18,7,1,1,4,5,62,2,1,...]</math>
|-
|<math>\arctan(1/7)</math><ref name=":2" />
|2
|4.076...
|<math>[0;7,21,8,1,3,1,8,2,6,1,...]</math>
|-
|<math>\arctan(1/10)</math><ref name=":2" />
|2
|4.595...
|<math>[0;10,30,12,1,1,7,3,2,1,3,...]</math>
|-
|<math>\arctan(1/4)</math><ref name=":2" />
|2
|5.793...
|<math>[0;4,12,5,12,1,1,1,3,2,1,...]</math>
| rowspan="3" |Of the form <math>\arctan(1/2^k)</math>
|-
|<math>\arctan(1/8)</math><ref name=":2" />
|2
|3.673...
|<math>[0;8,24,10,24,1,77,1,1,5,1,...]</math>
|-
|<math>\arctan(1/16)</math><ref name=":2" />
|2
|3.068...
|<math>[0;16,48,20,49,1,4,1,3,1,1,...]</math>
|-
|<math>\pi/\sqrt{3}</math><ref>{{Cite journal|last=Androsenko|first=V. A.|date=2015|title=<nowiki>Irrationality measure of the number \frac{\pi}{\sqrt{3}}</nowiki>|url=https://iopscience.iop.org/article/10.1070/IM2015v079n01ABEH002731|journal=Izvestiya: Mathematics|language=en|volume=79|issue=1|pages=1–17|doi=10.1070/im2015v079n01abeh002731|s2cid=123775303 |issn=1064-5632|via=}}</ref><ref name=":3">{{Cite journal|last=Polyanskii|first=A. A.|date=2018-03-01|title=On the Irrationality Measures of Certain Numbers. II|url=https://doi.org/10.1134/S0001434618030306|journal=Mathematical Notes|language=en|volume=103|issue=3|pages=626–634|doi=10.1134/S0001434618030306|s2cid=125251520|issn=1573-8876}}</ref>
|2
|4.60105...
|<math>[1;1,4,2,1,2,3,7,3,3,30,...]</math>
| rowspan="6" |Of the form <math>\sqrt{2k-1}\arctan\left({\frac{\sqrt{2k-1}}{k-1}}\right)</math>
|-
|<math>\sqrt{7}\arctan({\sqrt{7}/3})</math><ref name=":3" />
|2
|3.94704...
|<math>[1;1,10,2,1,1,2,3,6,1,3,...]</math>
|-
|<math>\sqrt{11}\arctan({\sqrt{11}/5})</math><ref name=":3" />
|2
|3.76069...
|<math>[1;1,16,2,1,1,3,1,6,1,24,...]</math>
|-
|<math>\sqrt{15}\arctan({\sqrt{15}/7})</math><ref name=":3" />
|2
|3.66666...
|<math>[1;1,22,2,1,1,5,2,3,10,2,...]</math>
|-
|<math>\sqrt{19}\arctan({\sqrt{19}/9})</math><ref name=":3" />
|2
|3.60809...
|<math>[1;1,28,2,1,1,6,1,72,2,1,...]</math>
|-
|<math>\sqrt{23}\arctan({\sqrt{23}/11})</math><ref name=":3" />
|2
|3.56730...
|<math>[1;1,34,2,1,1,8,1,1,5,2,...]</math>
|-
|<math>\sqrt{7}\ln\left(\frac{4+\sqrt{7}}{3}\right)</math><ref name=":3" />
|2
|6.64610... 
|<math>[2;9,1,1,2,2,8,2,1,3,2,...]</math>
| rowspan="9" |Of the form <math>\sqrt{2k+1}\ln\left(\frac{\sqrt{2k+1}+1}{\sqrt{2k+1}-1}\right)</math>
|-
|<math>\sqrt{11}\ln\left(\frac{6+\sqrt{11}}{5}\right)</math><ref name=":3" />
|2
|5.82337...
|<math>[2;15,1,1,2,3,1,2,10,1,4,...]</math>
|-
|<math>\sqrt{13}\ln\left(\frac{7+\sqrt{13}}{6}\right)</math><ref name=":3" />
|2
|3.51433...
|<math>[2;18,1,1,2,4,2,5,33,6,2,...]</math>
|-
|<math>\sqrt{15}\ln\left(\frac{8+\sqrt{15}}{7}\right)</math><ref name=":3" />
|2
|5.45248...
|<math>[2;21,1,1,2,5,4,3,2,1,1,...]</math>
|-
|<math>\sqrt{17}\ln\left(\frac{9+\sqrt{17}}{8}\right)</math><ref name=":3" />
|2
|3.47834...
|<math>[2;24,1,1,2,6,92,3,3,1,16,...]</math>
|-
|<math>\sqrt{19}\ln\left(\frac{10+\sqrt{19}}{9}\right)</math><ref name=":3" />
|2
|5.23162...
|<math>[2;27,1,1,2,6,1,3,1,2,1,...]</math>
|-
|<math>\sqrt{21}\ln\left(\frac{11+\sqrt{21}}{10}\right)</math><ref name=":3" />
|2
|3.45356...
|<math>[2;30,1,1,2,7,1,1,3,4,63,...]</math>
|-
|<math>\sqrt{23}\ln\left(\frac{12+\sqrt{23}}{11}\right)</math><ref name=":3" />
|2
|5.08120...
|<math>[2;33,1,1,2,8,2,2,1,9,4,...]</math>
|-
|<math>5\ln(3/2)</math><ref name=":3" />
|2
|3.43506...
|<math>[2;36,1,1,2,9,8,5,1,38,1,...]</math>
|-
|<math>\frac{\pi}{\sqrt{3}}\pm\ln(3)</math><ref name=":2" />
|2
|4.5586...
|<math>[2;1,10,2,2,1,1,17,1,4,1,...]</math> and <math>[0;1,2,1,1,22,14,3,1,1,1,...]</math>
|
|-
|<math>\sqrt{3}\ln(2+\sqrt{3})\pm\frac{\pi}{\sqrt{3}}</math><ref name=":2" />
|2
|6.1382...
|<math>[4;10,1,1,5,7,2,2,1,31,2,...]</math> and <math>[0;2,7,7,1,1,1,3,9,9,1,...]</math>
|
|-
|<math>\ln(5)+\frac{\pi}2</math><ref name=":2" />
|2
|59.976...
|<math>[3;5,1,1,4,1,2,19,1,3,...]</math>
|
|-
|<math>T_2(1/b), b\geq2</math><ref>{{Cite journal|last1=Adamczewski|first1=Boris|last2=Rivoal|first2=Tanguy|date=2009|title=Irrationality measures for some automatic real numbers|url=https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/irrationality-measures-for-some-automatic-real-numbers/F89F4B7BBC9A06B6E9934FB2C3AFFE4D|journal=Mathematical Proceedings of the Cambridge Philosophical Society|language=en|volume=147|issue=3|pages=659–678|doi=10.1017/S0305004109002643|bibcode=2009MPCPS.147..659A|s2cid=1689323|issn=1469-8064}}</ref>
|2
|4
|Infinite continued fraction.
|<math>T_2(1/b):=\sum_{n=1}^\infty t_nb^{n-1}</math> where <math>t_n</math> is the <math>n</math>-th term of the [[Thue–Morse sequence]].
|-
|[[Champernowne constant]]s <math>C_b</math> in base <math>b\geq2</math><ref>{{Cite journal|last=Amou|first=Masaaki|date=1991-02-01|title=Approximation to certain transcendental decimal fractions by algebraic numbers|journal=Journal of Number Theory|language=en|volume=37|issue=2|pages=231–241|doi=10.1016/S0022-314X(05)80039-3|issn=0022-314X|doi-access=free}}</ref>
| colspan="2" style="text-align: center;" |<math>b</math>
|Infinite continued fraction.
|Examples include <math>C_{10}=0.1234567891011...=[0;8,9,1,149083,1,...]</math>
|-
|Liouville numbers <math>L</math>
| colspan="2" style="text-align: center;" |<math>\infty</math>
|Infinite continued fraction, not behaving predictable.
|The Liouville numbers are precisely those numbers having infinite irrationality measure.<ref name="bugeaud" />{{rp|248}}
|}

===Irrationality base===

The ''irrationality base'' is a measure of irrationality introduced by J. Sondow<ref>{{cite arXiv |last=Sondow |first=Jonathan |year=2004 |title=Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik |eprint=math/0406300}}</ref> as an irrationality measure for Liouville numbers. It is defined as follows:

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

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

then <math>\beta</math> is called the irrationality base of <math>\alpha</math> and is represented as <math>\beta(\alpha)</math>

If no such <math>\beta</math> exists, then <math>\alpha</math> is called a ''super Liouville number''.

'''Example''': The series <math>\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</math> is a ''super Liouville number'', while the series <math>\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</math> is a Liouville number with irrationality base 2. (<math>{^{b}a}</math> represents [[Tetration|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|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 (mathematical constant)|''e'']], one can show that ''e'' is an example of a transcendental number that is not Liouville. [[Biography:Kurt Mahler|Mahler]] proved in 1953 that [[Pi|{{pi}}]] is another such example.<ref>Kurt Mahler, "On the approximation of π", ''Nederl. Akad. Wetensch. Proc. Ser. A.'', t.&nbsp;56 (1953), p.&nbsp;342–366.</ref>

The proof proceeds by first establishing a property of [[Irrational number|irrational]] [[Algebraic number|algebraic number]]s. 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 (mathematics)|lemma]] is usually known as '''Liouville's theorem (on diophantine approximation)''', there being several results known as Liouville's theorem<!--intentional link to DAB page-->.

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|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,

: <math>  \left| \alpha - \frac{p}{q}  \right | > \frac{A}{q^n} </math>

'''Proof of Lemma:''' Let ''M'' be the maximum value of |''f'' ′(''x'')| (the [[Absolute value|absolute value]] of the [[Derivative|derivative]] of ''f'') over the [[Interval (mathematics)|interval]] [''α''&nbsp;−&nbsp;1, ''α''&nbsp;+&nbsp;1]. Let ''α''<sub>1</sub>, ''α''<sub>2</sub>, ..., ''α''<sub>''m''</sub> be the distinct roots of ''f'' which differ from ''α''. Select some value ''A'' > 0 satisfying

: <math>A< \min \left(1, \frac{1}{M}, \left| \alpha - \alpha_1 \right|, \left| \alpha - \alpha_2 \right|, \ldots , \left| \alpha-\alpha_m \right| \right)  </math>

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

: <math>\left| \alpha - \frac{p}{q}\right| \le \frac{A}{q^n} \le A< \min\left(1, \frac{1}{M}, \left| \alpha - \alpha_1 \right|, \left|\alpha - \alpha_2 \right|, \ldots , \left| \alpha-\alpha_m \right| \right) </math>

Then ''p''/''q'' is in the interval [''α'' − 1, ''α'' + 1]; and ''p''/''q'' is not in {''α''<sub>1</sub>, ''α''<sub>2</sub>, ..., ''α''<sub>''m''</sub>}, 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|mean value theorem]], there exists an ''x''<sub>0</sub> between ''p''/''q'' and ''α'' such that

: <math>f(\alpha)-f(\tfrac{p}{q}) = (\alpha - \frac{p}{q}) \cdot f'(x_0)</math>

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

: <math>\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 |</math>

Now, ''f'' is of the form <math>\sum_{i=0}^n</math> ''c''<sub>''i''</sub> ''x''<sup>''i''</sup> where each ''c''<sub>''i''</sub> is an integer; so expressing  |''f''(''p''/''q'')| as

: <math>\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 | \ge \frac {1}{q^n} </math>

N.B. the last inequality holds because ''p''/''q'' is not a root of ''f'' and the ''c''<sub>''i''</sub> are integers.

Thus, |''f''(''p''/''q'')| ≥ 1/''q''<sup>''n''</sup>. Since |''f'' ′(''x''<sub>0</sub>)| ≤ ''M'' by the definition of ''M'', and 1/''M'' > ''A'' by the definition of ''A'', then

: <math>\left | \alpha - \frac{p}{q} \right | = \left|\frac{f(\tfrac{p}{q})}{f'(x_0)}\right| \ge \frac{1}{Mq^n} > \frac{A}{q^n} \ge \left| \alpha - \frac{p}{q} \right|</math>

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''

: <math>  \left| x - \frac{p}{q}  \right|> \frac{A}{q^{n}} </math>

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

: <math>\left|x-\frac ab\right|<\frac1{b^m}=\frac1{b^{r+n}}=\frac1{b^rb^n} \le \frac1{2^r}\frac1{b^n} \le \frac A{b^n} </math>

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

== See also ==
* [[Brjuno number]]
* [[Diophantine approximation]]

== References ==
{{reflist}}

==External links==
*[http://www.math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes5.pdf The Beginning of Transcendental Numbers]
{{Irrational number}}

{{DEFAULTSORT:Liouville Number}}
[[Category:Diophantine approximation]]
[[Category:Mathematical constants]]
[[Category:Articles containing proofs]]
[[Category:Real transcendental numbers]]
[[Category:Irrational numbers]]

{{Sourceattribution|Liouville number}}