Liouville Constant
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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, a Liouville number is a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
''x'' with the property that, for every positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''n'', there exists a pair of integers (''p, q'') with ''q'' > 1 such that :0 < \left, x-\frac\ < \frac . Liouville numbers are "almost
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
", and can thus be approximated "quite closely" by
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s of rational numbers. They are precisely those
transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
s that can be more closely approximated by rational numbers than any algebraic
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
. In 1844,
Joseph Liouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time. It is known that and are not Liouville numbers.


The existence of Liouville numbers (Liouville's constant)

Here we show that Liouville numbers exist by exhibiting a construction that produces such numbers. For any integer ''b'' ≥ 2 and any sequence of integers (''a''1, ''a''2, …, ) such that ''a''''k'' ∈  for all ''k'' and ''a''''k'' ≠ 0 for infinitely many ''k'', define the number :x = \sum_^\infty \frac. In the special case when ''b'' = 10, and ''a''''k'' = 1, for all ''k'', the resulting number ''x'' is called Liouville's constant: :''L'' = 0.11000100000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001... It follows from the definition of ''x'' that its base-''b'' representation is :x = \left(0.a_a_000a_00000000000000000a_0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000a_\ldots\right)_b\; where the ''n''th term is in the (''n''!)th place. Since this base-''b'' representation is non-repeating it follows that ''x'' is not a rational number. Therefore, for any rational number ''p''/''q'', we have , ''x'' − ''p''/''q'',  > 0. Now, for any integer ''n'' ≥ 1, define ''q''''n'' and ''p''''n'' as follows: :q_n = b^\,; \quad p_n = q_n \sum_^n \frac = \sum_^n \; . Then :\begin 0 < \left, x - \frac\ & = \left, x - \frac\ = \left, x - \sum_^n \frac\ = \left, \sum_^\infty \frac - \sum_^n \frac\ = \left, \left(\sum_^n \frac + \sum_^\infty \frac\right) - \sum_^n \frac\ = \sum_^\infty \frac \\ pt& \le \sum_^\infty \frac < \sum_^\infty \frac = \frac + \frac + \frac + ... = \frac + \frac + \frac + ... = \frac \sum_^\infty \frac \\ pt& = \frac\cdot\frac = \frac \le \frac = \frac = \frac = \frac = \frac = \frac \end Therefore, we conclude that any such ''x'' is a Liouville number.


Notes on the proof

# The inequality \sum_^\infty \frac \le \sum_^\infty \frac follows since ''a''''k'' ∈  for all ''k'', so at most ''a''''k'' = ''b''−1. The largest possible sum would occur if the sequence of integers (''a''1, ''a''2, …) were (''b''−1, ''b''−1, ...), i.e. ''a''''k'' = ''b''−1, for all ''k''. \sum_^\infty \frac will thus be less than or equal to this largest possible sum. # The strong inequality \begin \sum_^\infty \frac < \sum_^\infty \frac \end follows from our motivation to eliminate the
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...
by way of reducing it to a series for which we know a formula. In the proof so far, the purpose for introducing the inequality in 1. comes from intuition that \sum_^\infty \frac = \frac (the
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each succ ...
formula); therefore, if we can find an inequality from \sum_^\infty \frac that introduces a series with (''b''−1) in the numerator, and if we can work to further reduce the denominator term b^to b^, as well as shifting the series indices from 0 to \infty, then we will be able to eliminate both series and (''b''−1) terms, getting us closer to a fraction of the form \frac, which is the end-goal of the proof. We further this motivation here by selecting now from the sum \sum_^\infty \frac a partial sum. Observe that, for any term in \sum_^\infty \frac, since ''b'' ≥ 2, then \frac < \frac, for all ''k'' (except for when ''n''=1). Therefore, \begin \sum_^\infty \frac < \sum_^\infty \frac \end (since, even if ''n''=1, all subsequent terms are smaller). In order to manipulate the indices so that ''k'' starts at 0, we select a partial sum from within \sum_^\infty \frac (also less than the total value since it's a partial sum from a series whose terms are all positive). We will choose the partial sum formed by starting at ''k'' = (''n''+1)! which follows from our motivation to write a new series with ''k''=0, namely by noticing that b^ = b^b^0. #For the final inequality \frac \le \frac, we have chosen this particular inequality (true because ''b'' ≥ 2, where equality follows
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
''n''=1) because we wish to manipulate \frac into something of the form \frac. This particular inequality allows us to eliminate (''n''+1)! and the numerator, using the property that (''n''+1)! – ''n''! = (''n''!)''n'', thus putting the denominator in ideal form for the substitution q_n = b^.


Irrationality

Here we will show that the number ~ x = c / d ~, where and are integers and ~ d > 0 ~, cannot satisfy the inequalities that define a Liouville number. Since every
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
can be represented as such ~ c / d ~, we will have proven that no Liouville number can be rational. More specifically, we show that for any positive integer large enough that ~ 2^ > d > 0~ quivalently, for any positive integer ~ n > 1 + \log_2(d) ~) no pair of integers ~(\,p,\,q\,)~ exists that simultaneously satisfies the pair of bracketing inequalities :0 < \left, x - \frac\ < \frac~. If the claim is true, then the desired conclusion follows. Let and be any integers with ~q > 1~. Then we have, : \left, x - \frac \ = \left, \frac - \frac \ = \frac If \left, c\,q - d\,p \ = 0~, we would then have :\left, x - \frac\= \frac = 0 ~, meaning that such pair of integers ~(\,p,\,q\,)~ would violate the ''first'' inequality in the definition of a Liouville number, irrespective of any choice of  . If, on the other hand, since ~\left, c\,q - d\,p \ > 0 ~, then, since c\,q - d\,p is an integer, we can assert the sharper inequality \left, c\,q - d\,p \ \ge 1 ~. From this it follows that :\left, x - \frac\= \frac \ge \frac Now for any integer ~n > 1 + \log_2(d)~, the last inequality above implies :\left, x - \frac \ \ge \frac > \frac \ge \frac ~. Therefore, in the case ~ \left, c\,q - d\,p \ > 0 ~ such pair of integers ~(\,p,\,q\,)~ would violate the ''second'' inequality in the definition of a Liouville number, for some positive integer . We conclude that there is no pair of integers ~(\,p,\,q\,)~, with ~ q > 1 ~, that would qualify such an ~ 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 Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
.)


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 . 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 In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb ...
, the same thing occurs with the set of all Liouville numbers. Moreover, the Liouville numbers form a
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
subset of the set of real numbers.


Liouville numbers and measure

From the point of view of
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
, the set of all Liouville numbers ''L'' is small. More precisely, its
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
, λ(L), is zero. The proof given follows some ideas by John C. Oxtoby. For positive integers ''n'' > 2 and ''q'' ≥ 2 set: :V_=\bigcup\limits_^\infty \left(\frac-\frac,\frac+\frac\right) we have :L\subseteq \bigcup_^\infty V_. Observe that for each positive integer ''n'' ≥ 2 and ''m'' ≥ 1, we also have :L\cap (-m,m)\subseteq \bigcup\limits_^\infty V_\cap(-m,m)\subseteq \bigcup\limits_^\infty\bigcup\limits_^ \left( \frac-\frac,\frac+\frac\right). Since : \left, \left(\frac+\frac\right)-\left(\frac-\frac\right)\=\frac and ''n'' > 2 we have : \begin \mu(L\cap (-m,\, m)) & \leq\sum_^\infty\sum_^\frac = \sum_^\infty \frac \\ pt& \leq (4m+1)\sum_^\infty\frac \leq (4m+1) \int^\infty_1 \frac\leq\frac. \end Now :\lim_\frac=0 and it follows that for each positive integer ''m'', ''L'' ∩ (−''m'', ''m'') has Lebesgue measure zero. Consequently, so has ''L''. In contrast, the Lebesgue measure of the set of ''all'' real transcendental numbers is
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(since the set of algebraic numbers is a
null set In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null s ...
).


Structure of the set of Liouville numbers

For each positive integer , set :~ U_n = \bigcup\limits_^\infty ~ \bigcup\limits_^\infty ~ \left\ = \bigcup\limits_^\infty ~ \bigcup\limits_^\infty ~ \left(\frac-\frac~,~\frac + \frac\right) \setminus \left\ ~ The set of all Liouville numbers can thus be written as :~ L ~=~ \bigcap\limits_^\infty U_n ~=~ \bigcap\limits_ ~ \bigcup\limits_ ~ \bigcup \limits_\,\left(\,\left(\,\frac - \frac~,~ \frac + \frac \,\right) \setminus \left\ \,\right) ~. Each ~ U_n ~ is an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
; as its closure contains all rationals (the ~p / q~ from each punctured interval), it is also a
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
subset of real line. Since it is the intersection of countably many such open dense sets, is
comeagre In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called ...
, that is to say, it is a ''dense'' Gδ set.


Irrationality measure

The Liouville–Roth irrationality measure (irrationality exponent, approximation exponent, or Liouville–Roth constant) of a real number ''x'' is a measure of how "closely" it can be approximated by rationals. Generalizing the definition of Liouville numbers, instead of allowing any ''n'' in the power of ''q'', we find the largest possible value for ''μ'' such that 0< \left, x- \frac \ < \frac is satisfied by an infinite number of integer pairs (''p'', ''q'') with ''q'' > 0. This maximum value of ''μ'' is defined to be the irrationality measure of ''x''. For any value ''μ'' less than this upper bound, the infinite set of all rationals ''p''/''q'' satisfying the above inequality yield an approximation of ''x''. Conversely, if ''μ'' is greater than the upper bound, then there are at most finitely many (''p'', ''q'') with ''q'' > 0 that satisfy the inequality; thus, the opposite inequality holds for all larger values of ''q''. In other words, given the irrationality measure ''μ'' of a real number ''x'', whenever a rational approximation ''x'' ≅ ''p''/''q'', ''p'',''q'' ∈ N yields ''n'' + 1 exact decimal digits, we have :\frac \ge \left, x- \frac \ \ge \frac for any ε>0, except for at most a finite number of "lucky" pairs (''p'', ''q''). As a consequence of
Dirichlet's approximation theorem In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers \alpha and N , with 1 \leq N , there exist integers p and q such that 1 \leq q \leq N and ...
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. Below is a table of known upper and lower bounds for the irrationality measures of certain numbers.


Irrationality base

The ''irrationality base'' is a measure of irrationality introduced by J. Sondow as an irrationality measure for Liouville numbers. It is defined as follows: Let \alpha be an irrational number. If there exists a real number \beta \geq 1 with the property that for any \varepsilon >0 , there is a positive integer q(\varepsilon) such that : \left, \alpha-\frac \ > \frac 1 \text p,q \text q \geq q(\varepsilon) , then \beta is called the irrationality base of \alpha and is represented as \beta(\alpha) If no such \beta exists, then \alpha is called a ''super Liouville number''. Example: The series \varepsilon_=1+\frac+\frac+\frac+\frac+\frac+\ldots is a ''super Liouville number'', while the series \tau_2 = \sum_^\infty = \frac + \frac + \frac + \frac + \frac + \ldots is a Liouville number with irrationality base 2. ( represents
tetration In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common. Under the definition as rep ...
.)


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 In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
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 Gustav Mahler (; 7 July 1860 – 18 May 1911) was an Austro-Bohemian Romantic composer, and one of the leading conductors of his generation. As a composer he acted as a bridge between the 19th-century Austro-German tradition and the modernism ...
proved in 1953 that is another such example.The irrationality measure of does not exceed 7.6304, according to The proof proceeds by first establishing a property of
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
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 Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a ...
is usually known as Liouville's theorem (on diophantine approximation), there being several results known as Liouville's theorem. Below, we will show that no Liouville number can be algebraic. Lemma: If ''α'' is an irrational number which is the root of an irreducible
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
''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, : \left, \alpha - \frac \right , > \frac Proof of Lemma: Let ''M'' be the maximum value of , ''f'' ′(''x''), (the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of ''f'') over the interval 'α'' − 1, ''α'' + 1 Let ''α''1, ''α''2, ..., ''α''''m'' be the distinct roots of ''f'' which differ from ''α''. Select some value ''A'' > 0 satisfying : A< \min \left(1, \frac, \left, \alpha - \alpha_1 \, \left, \alpha - \alpha_2 \, \ldots , \left, \alpha-\alpha_m \ \right) Now assume that there exist some integers ''p'', ''q'' contradicting the lemma. Then : \left, \alpha - \frac\ \le \frac \le A< \min\left(1, \frac, \left, \alpha - \alpha_1 \, \left, \alpha - \alpha_2 \, \ldots , \left, \alpha-\alpha_m \ \right) Then ''p''/''q'' is in the interval 'α'' − 1, ''α'' + 1 and ''p''/''q'' is not in , 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 In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
, there exists an ''x''0 between ''p''/''q'' and ''α'' such that : f(\alpha)-f(\tfrac) = (\alpha - \frac) \cdot f'(x_0) Since ''α'' is a root of ''f'' but ''p''/''q'' is not, we see that , ''f'' ′(''x''0), > 0 and we can rearrange: : \left, \alpha -\frac\right , = \frac = \left , \frac \right , Now, ''f'' is of the form \sum_^n ''c''''i'' ''x''''i'' where each ''c''''i'' is an integer; so we can express , ''f''(''p''/''q''), as : \left, f \left (\frac \right) \ = \left, \sum_^n c_i p^i q^ \ = \frac \left, \sum_^n c_i p^i q^ \right , \ge \frac the last inequality holding because ''p''/''q'' is not a root of ''f'' and the ''c''''i'' are integers. Thus we have that , ''f''(''p''/''q''), ≥ 1/''q''''n''. Since , ''f'' ′(''x''0), ≤ ''M'' by the definition of ''M'', and 1/''M'' > ''A'' by the definition of ''A'', we have that : \left , \alpha - \frac \right , = \left, \frac\ \ge \frac > \frac \ge \left, \alpha - \frac \ 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'' : \left, x - \frac \> \frac Let ''r'' be a positive integer such that 1/(2''r'') ≤ ''A''. If we let ''m'' = ''r'' + ''n'', and since ''x'' is a Liouville number, then there exist integers ''a'', ''b'' where ''b'' > 1 such that : \left, x-\frac ab\<\frac1=\frac1=\frac1 \le \frac1\frac1 \le \frac A which contradicts the lemma. Hence, if a Liouville number exists, it cannot be algebraic, and therefore must be transcendental.


See also

*
Brjuno number In mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of irrational number named for Russian mathematician Alexander Bruno, who introduced them in . Formal definition An irrational number \alpha is called a Brju ...
*
Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by r ...


References


External links


The Beginning of Transcendental Numbers
{{DEFAULTSORT:Liouville Number Diophantine approximation Mathematical constants Articles containing proofs Real transcendental numbers Irrational numbers