Inmathematics, anirrationality measure of areal number${\displaystyle x}$ is a measure of how "closely" it can beapproximated byrationals.

If afunction${\displaystyle f(t,\lambda )}$, defined for${\displaystyle t,\lambda >0}$, takes positive real values and is strictly decreasing in both variables, consider the followinginequality:

for a given real number${\displaystyle x\in \mathbb{R}}$ and rational numbers${\displaystyle \frac{p}{q}}$ with${\displaystyle p\in \mathbb{Z},q\in {\mathbb{Z}}^{+}}$. Define${\displaystyle R}$ as theset of all${\displaystyle \lambda \in {\mathbb{R}}^{+}}$ for which only finitely many${\displaystyle \frac{p}{q}}$ exist, such that the inequality is satisfied. Then${\displaystyle \lambda (x)=infR}$ is called an irrationality measure of${\displaystyle x}$ with regard to${\displaystyle f.}$ If there is no such${\displaystyle \lambda }$ and the set${\displaystyle R}$ isempty,${\displaystyle x}$ is said to have infinite irrationality measure${\displaystyle \lambda (x)=\mathrm{\infty }}$.

Consequently, the inequality

has at most only finitely many solutions${\displaystyle \frac{p}{q}}$ for all${\displaystyle \epsilon >0}$.^[1]

Theirrationality exponent orLiouville–Roth irrationality measure is given by setting${\displaystyle f(q,\mu )=q^{-\mu }}$,^[1] a definition adapting the one ofLiouville numbers — the irrationality exponent${\displaystyle \mu (x)}$ is defined for real numbers${\displaystyle x}$ to be thesupremum of the set of${\displaystyle \mu }$ such that is satisfied by an infinite number ofcoprime integer pairs${\displaystyle (p,q)}$ with${\displaystyle q>0}$.^{[2][3]: 246}

For any value, theinfinite set of all rationals${\displaystyle p/q}$ satisfying the above inequality yields good approximations of${\displaystyle x}$. Conversely, if${\displaystyle n>\mu (x)}$, then there are at most finitely many coprime${\displaystyle (p,q)}$ with${\displaystyle q>0}$ that satisfy the inequality.

For example, whenever a rational approximation${\displaystyle \frac{p}{q}\approx x}$ with${\displaystyle p,q\in \mathbb{N}}$ yields${\displaystyle n+1}$ exact decimal digits, then

${\displaystyle \frac{1}{{10}^n}\ge \left|x-\frac{p}{q}\right|\ge \frac{1}{q^{\mu (x)+\epsilon }}}$

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

A number${\displaystyle x\in \mathbb{R}}$ with irrationality exponent${\displaystyle \mu (x)\le 2}$ is called adiophantine number,^[4] while numbers with${\displaystyle \mu (x)=\mathrm{\infty }}$ are calledLiouville numbers.

Rational numbers have irrationality exponent 1, while (as a consequence ofDirichlet's approximation theorem) every irrational number has irrationality exponent at least 2.

On the other hand, an application ofBorel-Cantelli lemma shows that almost all numbers, including allalgebraic irrational numbers, have an irrationality exponent exactly equal to 2.^[3]: 246

It is${\displaystyle \mu (x)=\mu (rx+s)}$ for real numbers${\displaystyle x}$ and rational numbers${\displaystyle r\ne 0}$ and${\displaystyle s}$. If for some${\displaystyle x}$ we have${\displaystyle \mu (x)\le \mu }$, then it follows${\displaystyle \mu (x^{1/2})\le 2\mu }$.^[5]: 368

For a real number${\displaystyle x}$ given by itssimple continued fraction expansion${\displaystyle x=[a_0;a_1,a_2,...]}$ with convergents${\displaystyle p_i/q_i}$ it holds:^[1]

${\displaystyle \mu (x)=1+\underset{n\to \mathrm{\infty }}{lim sup}\frac{\ln q_{n+1}}{\ln q_n}=2+\underset{n\to \mathrm{\infty }}{lim sup}\frac{\ln a_{n+1}}{\ln q_n}.}$

If we have${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}\frac{1}{n}\ln |q_n|\le \sigma }$ and${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\ln |q_{n}x-p_n|=-\tau }$ for some positive real numbers${\displaystyle \sigma ,\tau }$, then we can establish an upper bound for the irrationality exponent of${\displaystyle x}$ by:^[6][7]

${\displaystyle \mu (x)\le 1+\frac{\sigma }{\tau }}$

For mosttranscendental numbers, the exact value of their irrationality exponent is not known.^[5] Below is a table of knownupper and lower bounds.

Number${\displaystyle x}$	Irrationality exponent${\displaystyle \mu (x)}$		Notes
	Lower bound	Upper bound
Rational number${\displaystyle p/q}$ with${\displaystyle p\in \mathbb{Z},q\in {\mathbb{Z}}^{+}}$	1		Every rational number${\displaystyle p/q}$ has an irrationality exponent of exactly 1.
Irrationalalgebraic number${\displaystyle \alpha }$	2		ByRoth's theorem the irrationality exponent of any irrational algebraic number is exactly 2. Examples includesquare roots and thegolden ratio${\displaystyle \phi }$.
${\displaystyle e^{2/k},k\in {\mathbb{Z}}^{+}}$	2		If the elements${\displaystyle a_n}$ of the simple continued fraction expansion of an irrational number${\displaystyle x}$ are bounded above by an arbitrary polynomial${\displaystyle P}$, then its irrationality exponent is${\displaystyle \mu (x)=2}$. Examples include numbers whose continued fractions behave predictably such as ${\displaystyle e=[2;1,2,1,1,4,1,1,6,1,...]}$ and${\displaystyle I_0(2)/I_1(2)=[1;2,3,4,5,6,7,8,9,10,...]}$.
${\displaystyle \tan (1/k),k\in {\mathbb{Z}}^{+}}$	2
${\displaystyle \tanh (1/k),k\in {\mathbb{Z}}^{+}}$	2
${\displaystyle S(b)}$ with${\displaystyle b\ge 2}$	2		${\displaystyle S(b):=\sum _{k=0}^{\mathrm{\infty }}{b}^{-2^k}}$with${\displaystyle b\in \mathbb{Z}}$, has continued fraction terms which do not exceed a fixed constant.[8][9]
${\displaystyle T(b)}$ with${\displaystyle b\ge 2}$[10]	2		${\displaystyle T(b):=\sum _{k=0}^{\mathrm{\infty }}{t}_k{b}^{-k}}$ where${\displaystyle t_k}$ is theThue–Morse sequence and${\displaystyle b\in \mathbb{Z}}$. SeeProuhet-Thue-Morse constant.
${\displaystyle \ln (2)}$[11][12]	2	3.57455...	There are other numbers of the form${\displaystyle \ln (a/b)}$ for which bounds on their irrationality exponents are known.[13][14][15]
${\displaystyle \ln (3)}$[11][16]	2	5.11620...
${\displaystyle 5\ln (3/2)}$[17]	2	3.43506...	There are many other numbers of the form${\displaystyle \sqrt{2k+1}\ln \left(\frac{\sqrt{2k+1}+1}{\sqrt{2k+1}-1}\right)}$ for which bounds on their irrationality exponents are known.[17] This is the case for${\displaystyle k=12}$.
${\displaystyle \pi /\sqrt{3}}$[18][19]	2	4.60105...	There are many other numbers of the form${\displaystyle \sqrt{2k-1}\arctan \left(\frac{\sqrt{2k-1}}{k-1}\right)}$ for which bounds on their irrationality exponents are known.[18] This is the case for${\displaystyle k=2}$.
${\displaystyle \pi }$[11][20]	2	7.10320...	It has been proven that if theFlint Hills series${\displaystyle {\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{{\csc }^{2}n}{n^3}}}$ (wheren is in radians) converges, then${\displaystyle \pi }$'s irrationality exponent is at most${\displaystyle 5/2}$[21][22] and that if it diverges, the irrationality exponent is at least${\displaystyle 5/2}$.[23]
${\displaystyle {\pi }^2}$[11][24]	2	5.09541...	${\displaystyle {\pi }^2}$ and${\displaystyle \zeta (2)}$ are linearly dependent over${\displaystyle \mathbb{Q}}$.${\displaystyle \left(\zeta (2)=\frac{{\pi }^2}{6}\right)}$, also see theBasel problem.
${\displaystyle \arctan (1/2)}$[25]	2	9.27204...	There are many other numbers of the form${\displaystyle \arctan (1/k)}$ for which bounds on their irrationality exponents are known.[26][27]
${\displaystyle \arctan (1/3)}$[28]	2	5.94202...
Apéry's constant${\displaystyle \zeta (3)}$[11]	2	5.51389...
${\displaystyle \mathrm{\Gamma }(1/4)}$[29]	2	10330
Cahen's constant${\displaystyle C}$[30]	3
Champernowne constants${\displaystyle C_b}$ in base${\displaystyle b\ge 2}$[31]	${\displaystyle b}$		Examples include${\displaystyle C_{10}=\mathrm{0.1234567891011...}=[0;8,9,1,149083,1,...]}$
Liouville numbers${\displaystyle L}$	${\displaystyle \mathrm{\infty }}$		The Liouville numbers are precisely those numbers having infinite irrationality exponent.[3]: 248

Theirrationality base orSondow irrationality measure is obtained by setting${\displaystyle f(q,\beta )={\beta }^{-q}}$.^[1][6] It is a weaker irrationality measure, being able to distinguish how well different Liouville numbers can be approximated, but yielding${\displaystyle \beta (x)=1}$ for all other real numbers:

Let${\displaystyle x}$ be an irrational number. If there exist real numbers${\displaystyle \beta \ge 1}$ with the property that for any${\displaystyle \epsilon >0}$, there is a positive integer${\displaystyle q(\epsilon )}$ such that

${\displaystyle \left|x-\frac{p}{q}\right|>\frac{1}{(\beta +\epsilon {)}^q}}$

for all integers${\displaystyle p,q}$ with${\displaystyle q\ge q(\epsilon )}$ then the least such${\displaystyle \beta }$ is called the irrationality base of${\displaystyle x}$ and is represented as${\displaystyle \beta (x)}$.

If no such${\displaystyle \beta }$ exists, then${\displaystyle \beta (x)=\mathrm{\infty }}$ and${\displaystyle x}$ is called asuper Liouville number.

If a real number${\displaystyle x}$ is given by itssimple continued fraction expansion${\displaystyle x=[a_0;a_1,a_2,...]}$ with convergents${\displaystyle p_i/q_i}$ then it holds:

${\displaystyle \beta (x)=\underset{n\to \mathrm{\infty }}{lim sup}\frac{\ln q_{n+1}}{q_n}=\underset{n\to \mathrm{\infty }}{lim sup}\frac{\ln a_{n+1}}{q_n}}$.^[1]

Any real number${\displaystyle x}$ with finite irrationality exponent has irrationality base${\displaystyle \beta (x)=1}$, while any number with irrationality base${\displaystyle \beta (x)>1}$ has irrationality exponent${\displaystyle \mu (x)=\mathrm{\infty }}$ and is a Liouville number.

The number${\displaystyle L=[1;2,2^2,2^{2^2},...]}$ has irrationality exponent${\displaystyle \mu (L)=\mathrm{\infty }}$ and irrationality base${\displaystyle \beta (L)=1}$.

The numbers${\displaystyle {\tau }_a=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{}^{n}a}=1+\frac{1}{a}+\frac{1}{a^a}+\frac{1}{a^{a^a}}+\frac{1}{a^{a^{a^a}}}+...}$ (${\displaystyle {}^{n}a}$ representstetration,${\displaystyle a=2,3,\mathrm{4...}}$) have irrationality base${\displaystyle \beta ({\tau }_a)=a}$.

The number${\displaystyle S=1+\frac{1}{2^1}+\frac{1}{4^{2^1}}+\frac{1}{8^{4^{2^1}}}+\frac{1}{{16}^{8^{4^{2^1}}}}+\frac{1}{{32}^{{16}^{8^{4^{2^1}}}}}+\dots }$ has irrationality base${\displaystyle \beta (S)=\mathrm{\infty }}$, hence it is asuper Liouville number.

Although it is not known whether or not${\displaystyle e^{\pi }}$ is a Liouville number,^[32]: 20 it is known that${\displaystyle \beta (e^{\pi })=1}$.^[5]: 371

Setting${\displaystyle f(q,M)=(M{q}^2{)}^{-1}}$ gives a stronger irrationality measure: theMarkov constant${\displaystyle M(x)}$. For an irrational number${\displaystyle x\in \mathbb{R}\setminus \mathbb{Q}}$ it is the factor by whichDirichlet's approximation theorem can be improved for${\displaystyle x}$. Namely if is a positive real number, then the inequality

has infinitely many solutions${\displaystyle \frac{p}{q}\in \mathbb{Q}}$. If${\displaystyle c>M(x)}$ there are at most finitely many solutions.

Dirichlet's approximation theorem implies${\displaystyle M(x)\ge 1}$ andHurwitz's theorem gives${\displaystyle M(x)\ge \sqrt{5}}$ both for irrational${\displaystyle x}$.^[33]

This is in fact the best general lower bound since thegolden ratio gives${\displaystyle M(\phi )=\sqrt{5}}$. It is also${\displaystyle M(\sqrt{2})=2\sqrt{2}}$.

Given${\displaystyle x=[a_0;a_1,a_2,...]}$ by its simplecontinued fraction expansion, one may obtain:^[34]

${\displaystyle M(x)=\underset{n\to \mathrm{\infty }}{lim sup}([a_{n+1};a_{n+2},a_{n+3},...]+[0;a_n,a_{n-1},...,a_2,a_1]).}$

Bounds for the Markov constant of${\displaystyle x=[a_0;a_1,a_2,...]}$ can also be given by with${\displaystyle p=\underset{n\to \mathrm{\infty }}{lim sup}{a}_n}$.^[35] This implies that${\displaystyle M(x)=\mathrm{\infty }}$ if and only if${\displaystyle (a_k)}$ is not bounded. The numbers with finite${\displaystyle M(x)}$ are calledbadly approximable. In particular everyquadratic irrational number${\displaystyle x}$ is badly approximable.

Any number with${\displaystyle \mu (x)>2}$ or${\displaystyle \beta (x)>1}$ has an unbounded simple continued fraction and hence${\displaystyle M(x)=\mathrm{\infty }}$. A further consequence is${\displaystyle M(e)=\mathrm{\infty }}$.

For rational numbers${\displaystyle r}$ it may be defined${\displaystyle M(r)=0}$.

The values${\displaystyle M(e)=\mathrm{\infty }}$ and${\displaystyle \mu (e)=2}$ imply that the inequality has for all${\displaystyle c\in {\mathbb{R}}^{+}}$ infinitely many solutions${\displaystyle \frac{p}{q}\in \mathbb{Q}}$ while the inequality has for all${\displaystyle \epsilon \in {\mathbb{R}}^{+}}$ only at most finitely many solutions${\displaystyle \frac{p}{q}\in \mathbb{Q}}$ . This gives rise to the question what the best upper bound is. The answer is given by:^[36]

which is satisfied by infinitely many${\displaystyle \frac{p}{q}\in \mathbb{Q}}$ for${\displaystyle c>\frac{1}{2}}$ but not for.

This makesthe number${\displaystyle e}$ alongside the rationals and quadratic irrationals an exception to the fact that for almost all real numbers${\displaystyle x\in \mathbb{R}}$ the inequality below has infinitely many solutions${\displaystyle \frac{p}{q}\in \mathbb{Q}}$:^[5] (seeKhinchin's theorem)

Kurt Mahler extended the concept of an irrationality measure and defined a so-calledtranscendence measure, drawing on the idea of a Liouville number andpartitioning the transcendental numbers into three distinct classes.^[3]

Instead of taking for a given real number${\displaystyle x}$ the difference${\displaystyle |x-p/q|}$ with${\displaystyle p/q\in \mathbb{Q}}$, one may instead focus on term${\displaystyle |qx-p|=|L(x)|}$ with${\displaystyle p,q\in \mathbb{Z}}$ and${\displaystyle L\in \mathbb{Z}[x]}$ with${\displaystyle \mathrm{deg}L=1}$. Consider the following inequality:

with${\displaystyle p,q\in \mathbb{Z}}$ and${\displaystyle \omega \in {\mathbb{R}}_0^{+}}$.

Define${\displaystyle R}$ as the set of all${\displaystyle \omega \in {\mathbb{R}}_0^{+}}$ for which infinitely many solutions${\displaystyle p,q\in \mathbb{Z}}$ exist, such that the inequality is satisfied. Then${\displaystyle {\omega }_1(x)=supM}$ is Mahler's irrationality measure. It gives${\displaystyle {\omega }_1(p/q)=0}$ for rational numbers,${\displaystyle {\omega }_1(\alpha )=1}$ for algebraic irrational numbers and in general${\displaystyle {\omega }_1(x)=\mu (x)-1}$, where${\displaystyle \mu (x)}$ denotes the irrationality exponent.

Mahler's irrationality measure can be generalized as follows:^[2][3] Take${\displaystyle P}$ to be a polynomial with${\displaystyle \mathrm{deg}P\le n\in {\mathbb{Z}}^{+}}$ and integer coefficients${\displaystyle a_i\in \mathbb{Z}}$. Then define aheight function${\displaystyle H(P)=max(|a_0|,|a_1|,...,|a_n|)}$ and consider forcomplex numbers${\displaystyle z}$ the inequality:

with${\displaystyle \omega \in {\mathbb{R}}_0^{+}}$.

Set${\displaystyle R}$ to be the set of all${\displaystyle \omega \in {\mathbb{R}}_0^{+}}$ for which infinitely many such polynomials exist, that keep the inequality satisfied. Further define${\displaystyle {\omega }_n(z)=supR}$ for all${\displaystyle n\in {\mathbb{Z}}^{+}}$ with${\displaystyle {\omega }_1(z)}$ being the above irrationality measure,${\displaystyle {\omega }_2(z)}$ being anon-quadraticity measure, etc.

Then Mahler's transcendence measure is given by:

${\displaystyle \omega (z)=\underset{n\to \mathrm{\infty }}{lim sup}{\omega }_n(z).}$

The transcendental numbers can now be divided into the following three classes:

If for all${\displaystyle n\in {\mathbb{Z}}^{+}}$ the value of${\displaystyle {\omega }_n(z)}$ is finite and${\displaystyle \omega (z)}$ is finite as well,${\displaystyle z}$ is called anS-number (of type${\displaystyle \omega (z)}$).

If for all${\displaystyle n\in {\mathbb{Z}}^{+}}$ the value of${\displaystyle {\omega }_n(z)}$ is finite but${\displaystyle \omega (z)}$ is infinite,${\displaystyle z}$ is called anT-number.

If there exists a smallest positive integer${\displaystyle N}$ such that for all${\displaystyle n\ge N}$ the${\displaystyle {\omega }_n(z)}$ are infinite,${\displaystyle z}$ is called anU-number (of degree${\displaystyle N}$).

The number${\displaystyle z}$ is algebraic (and called anA-number) if and only if${\displaystyle \omega (z)=0}$.

Almost all numbers are S-numbers. In fact, almost all real numbers give${\displaystyle \omega (x)=1}$ while almost all complex numbers give${\displaystyle \omega (z)=\frac{1}{2}}$.^[37]: 86 The numbere is an S-number with${\displaystyle \omega (e)=1}$. The numberπ is either an S- or T-number.^[37]: 86 The U-numbers are a set of measure 0 but still uncountable.^[38] They contain the Liouville numbers which are exactly the U-numbers of degree one.

Another generalization of Mahler's irrationality measure gives a linear independence measure.^[2][13] For real numbers${\displaystyle x_1,...,x_n\in \mathbb{R}}$ consider the inequality

with${\displaystyle c_1,...,c_n\in \mathbb{Z}}$ and${\displaystyle \nu \in {\mathbb{R}}_0^{+}}$.

Define${\displaystyle R}$ as the set of all${\displaystyle \nu \in {\mathbb{R}}_0^{+}}$ for which infinitely many solutions${\displaystyle c_1,...c_n\in \mathbb{Z}}$ exist, such that the inequality is satisfied. Then${\displaystyle \nu (x_1,...,x_n)=supR}$ is the linear independence measure.

If the${\displaystyle x_1,...,x_n}$ are linearly dependent over${\displaystyle \mathbb{Q}}$ then${\displaystyle \nu (x_1,...,x_n)=0}$.

If${\displaystyle 1,x_1,...,x_n}$ arelinearly independent algebraic numbers over${\displaystyle \mathbb{Q}}$ then${\displaystyle \nu (1,x_1,...,x_n)\le n}$.^[32]

It is further${\displaystyle \nu (1,x)={\omega }_1(x)=\mu (x)-1}$.

Jurjen Koksma in 1939 proposed another generalization, similar to that of Mahler, based on approximations of complex numbers by algebraic numbers.^[3][37]

For a given complex number${\displaystyle z}$ consider algebraic numbers${\displaystyle \alpha }$ of degree at most${\displaystyle n}$. Define a height function${\displaystyle H(\alpha )=H(P)}$, where${\displaystyle P}$ is thecharacteristic polynomial of${\displaystyle \alpha }$ and consider the inequality:

with${\displaystyle {\omega }^{\ast }\in {\mathbb{R}}_0^{+}}$.

Set${\displaystyle R}$ to be the set of all${\displaystyle {\omega }^{\ast }\in {\mathbb{R}}_0^{+}}$ for which infinitely many such algebraic numbers${\displaystyle \alpha }$ exist, that keep the inequality satisfied. Further define${\displaystyle {\omega }_n^{\ast }(z)=supR}$ for all${\displaystyle n\in {\mathbb{Z}}^{+}}$ with${\displaystyle {\omega }_1^{\ast }(z)}$ being an irrationality measure,${\displaystyle {\omega }_2^{\ast }(z)}$ being anon-quadraticity measure,^[17] etc.

Then Koksma's transcendence measure is given by:

${\displaystyle {\omega }^{\ast }(z)=\underset{n\to \mathrm{\infty }}{lim sup}{\omega }_n^{\ast }(z)}$.

The complex numbers can now once again be partitioned into four classes A*, S*, T* and U*. However it turns out that these classes are equivalent to the ones given by Mahler in the sense that they produce exactly the same partition.^[37]: 87

Given a real number${\displaystyle x\in \mathbb{R}}$, an irrationality measure of${\displaystyle x}$ quantifies how well it can be approximated by rational numbers${\displaystyle \frac{p}{q}}$ with denominator${\displaystyle q\in {\mathbb{Z}}^{+}}$. If${\displaystyle x=\alpha }$ is taken to be an algebraic number that is also irrational one may obtain that the inequality

has only at most finitely many solutions${\displaystyle \frac{p}{q}\in \mathbb{Q}}$ for${\displaystyle \mu >2}$. This is known asRoth's theorem.

This can be generalized: Given a set of real numbers${\displaystyle x_1,...,x_n\in \mathbb{R}}$ one can quantify how well they can be approximated simultaneously by rational numbers${\displaystyle \frac{p_1}{q},...,\frac{p_n}{q}}$ with the same denominator${\displaystyle q\in {\mathbb{Z}}^{+}}$. If the${\displaystyle x_i={\alpha }_i}$ are taken to be algebraic numbers, such that${\displaystyle 1,{\alpha }_1,...,{\alpha }_n}$ are linearly independent over the rational numbers${\displaystyle \mathbb{Q}}$ it follows that the inequalities

have only at most finitely many solutions${\displaystyle \left(\frac{p_1}{q},...,\frac{p_n}{q}\right)\in {\mathbb{Q}}^n}$ for${\displaystyle \mu >1+\frac{1}{n}}$. This result is due toWolfgang M. Schmidt.^[39][40]

• Diophantine approximation
• Transcendental number
• Continued fraction
• Brjuno number

• Diophantine approximation
• Transcendental numbers
• Mathematical constants
• Irrational numbers

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