In
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
, a
Liouville number is a
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
x with the property that, for any positive
integerThe integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
n, there exist integers
p and
q with
q > 1 and such that
A Liouville number can thus be approximated "quite closely" by a
sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence...
of rational numbers. In 1844,
Joseph LiouvilleJoseph Liouville was a French mathematician.- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...
showed that all Liouville numbers are
transcendentalIn mathematics, a transcendental number is a number that is not algebraic, that is, not a solution of a non-constant polynomial equation with rational coefficients....
, thus establishing the existence of transcendental numbers for the first time.
Elementary properties
An equivalent definition to the one given above is that for any positive integer
n, there exists an
infinite number of pairs of integers (
p,
q) obeying the above inequality.
It is relatively easily proven that if
x is a Liouville number,
x is
irrationalIn mathematics, an irrational number is any real number that is not a rational number—that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Informally, this means numbers that cannot be represented as simple fractions...
. Assume otherwise; then there exists integers
c,
d with
x =
c/
d. Let
n be a positive integer such that 2
n − 1 >
d. Then if
p and
q are any integers such that
q > 1 and
p/
q ≠
c/
d, then
which contradicts the definition of Liouville number.
Liouville constant
The number
is known as
Liouville's constant. Liouville's constant is a Liouville number; if we define
pn and
qn as follows:
then we have for all positive integers
n
Uncountability
Consider, for example, the number
3.140001000000000000000005....
where a digit is zero except in position n! where the digit equals the nth digit following the decimal point in the decimal expansion of π.
This number, as well as any other decimal with its non-zero digits similarly situated, satisfies the definition of Liouville number.
Cantor's second diagonal argumentCantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers...
shows that the set of such numbers is uncountable. Moreover, the Liouville numbers form an uncountable set of measure zero, dense in the set of real numbers.
Liouville numbers and measure
From the point of view of measure theory, the set of all Liouville numbers is small. More precisely, its
Lebesgue measureIn mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration...
is zero. The proof given follows some ideas by John C. Oxtoby.
For positive integers and set: – we have
Observe that for each positive integer and , we also have
Since and we have
Now 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
T of
all real transcendental numbers is infinite (since
T is the complement of a null set).
Liouville numbers and topology
For each positive integer
n, set.
The set of all Liouville numbers can thus be written as .
Each is an
open setIn mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...
; as its closure contains all rationals (the {p/q}'s from each punctured interval), it is also a
denseIn topology and related areas of mathematics, a subset A of a topological space X is called dense if any point in X can be "well-approximated" by points in A...
subset of real line. Since it is the intersection of countably many such open dense sets, is
comeagreIn the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible...
, that is to say, it is a
dense GδIn the mathematical field of topology, a Gδ set, is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt .The term inner limiting set is also used...
set.
Irrationality measure
The
irrationality measure (or
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 least upper bound of the set of
real numbers μ such that
is satisfied by an infinite number of integer pairs (
p,
q) with
q > 0. This least upper bound 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
except for at most a finite number of “lucky” pairs (
p,
q).
For a rational number
α the irrationality measure is μ(
α) = 1.
The
Thue–Siegel–Roth theoremIn mathematics, the Thue–Siegel–Roth theorem, also known simply as Roth's theorem, is a foundational result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that a given algebraic number α may not have too many rational number approximations, that are 'very...
proves that if
α is an
algebraic numberIn mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational coefficients...
, real but not rational, it is μ(
α) = 2.
Transcendental numbers have irrationality measure 2 or greater. As an example,
eThe mathematical constant e is the unique real number such that the value of the derivative of the function f = e
x at the point x = 0 is exactly 1. The function e
x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm...
has μ(
e) = 2 even though
e is transcendental.
The Liouville numbers are precisely those numbers having infinite irrationality measure.
Liouville numbers and transcendency
All Liouville numbers are
transcendentalIn mathematics, a transcendental number is a number that is not algebraic, that is, not a solution of a non-constant polynomial equation with rational coefficients....
, as will be proven below. Establishing that a given number is a Liouville number provides a useful tool for proving a given number is transcendental. Unfortunately, not every transcendental number is a Liouville number. The terms in the
continued fractionIn mathematics, a continued fraction is an expression such aswhere a0 is an integer and all the other numbers ai are positive integers. Longer expressions are defined analogously...
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
eThe mathematical constant e is the unique real number such that the value of the derivative of the function f = e
x at the point x = 0 is exactly 1. The function e
x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm...
, one can show that
e is an example of a transcendental number that is not Liouville. Mahler proved in 1953 that
πPi or π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. The symbol π was first proposed by the Welsh mathematician William Jones in 1706...
is another such example.
The proof proceeds by first establishing a property of
irrationalIn mathematics, an irrational number is any real number that is not a rational number—that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Informally, this means numbers that cannot be represented as simple fractions...
algebraic numberIn mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational coefficients...
s. This property essentially says that irrational algebraic numbers cannot be well approximated by rational numbers. A Liouville number is irrational but does not have this property, so it can't be algebraic and must be transcendental. The following
lemmaIn mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself...
is usually known as
Liouville's theorem (on diophantine approximation), there being several results known as
Liouville's theoremLiouville's theorem has various meanings, all mathematical results named after Joseph Liouville:*In complex analysis, see Liouville's theorem .*In conformal mappings, see Liouville's theorem ....
.
Lemma: If α is an irrational number which is the root of a
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
f of
degreeIn mathematics, there are several meanings of degree depending on the subject.- Unit of angle :A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation...
n > 0 with integer coefficients, then there exists a real number
A > 0 such that, for all integers
p,
q, with
q > 0,
Proof of Lemma: Let
M be the maximum value of |
f ′(
x)| (the
absolute valueIn mathematics, the absolute value of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.The absolute value of a number is denoted by ....
of the
derivativeIn calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
of
f) over the
intervalIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
[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
Now assume that there exists some integers
p,
q contradicting the lemma. Then
Then
p/
q is in the interval [α − 1, α + 1]; and
p/
q is not in {α
1, α
2, ..., 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 theoremIn calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is at least one point on that section at which the derivative of the curve is equal to the "average" derivative of the section...
, there exists an
x0 between
p/
q and α such that
Since α is a root of
f but
p/
q is not, we see that |
f ′(
x0)| > 0 and we can rearrange:
Now,
f is of the form ∑
i = 1 to n ci xi where each
ci is an integer; so we can express |
f(
p/
q)| as
the last inequality holding because
p/
q is not a root of
f and the
ci are integers.
Thus we have that |
f(
p/
q)| ≥ 1/
qn. Since |
f ′(
x0)| ≤
M by the definition of
M, and 1/
M >
A by the definition of
A, we have that
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
Let
r be a positive integer such that 1/(2
r) ≤
A. If we let
m =
r +
n, then, since
x is a Liouville number, there exists integers
a,
b > 1 such that
which contradicts the lemma; therefore
x is not algebraic, and is thus transcendental.
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