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In number theory, a **Liouville number** is a real number with the property that, for every positive integer , there exists a pair of integers with such that

The inequality implies that Liouville numbers possess an excellent sequence of rational number approximations. In 1844, Joseph Liouville proved a bound showing that there is a limit to how well algebraic numbers can be approximated by rational numbers, and he defined Liouville numbers specifically so that they would have rational approximations better than the ones allowed by this bound. Liouville also exhibited examples of Liouville numbers^{[1]} thereby establishing the existence of transcendental numbers for the first time.^{[2]}
One of these examples is **Liouville's constant**

in which the *n*th digit after the decimal point is 1 if is the factorial of a positive integer and 0 otherwise. It is known that π and e, although transcendental, are not Liouville numbers.^{[3]}

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

For any integer and any sequence of integers such that for all and for infinitely many , define the number

In the special case when , and for all , the resulting number is called Liouville's constant:

It follows from the definition of that its base- representation is

where the th term is in the th place.

Since this base- representation is non-repeating it follows that is not a rational number. Therefore, for any rational number , .

Now, for any integer , and can be defined as follows:

Then,

Therefore, any such is a Liouville number.

- The inequality follows since
*a*_{k}∈ {0, 1, 2, ...,*b*−1} for all*k*, so at most*a*_{k}=*b*−1. The largest possible sum would occur if the sequence of integers (*a*_{1},*a*_{2}, ...) were (*b*−1,*b*−1, ...), i.e.*a*_{k}=*b*−1, for all*k*. will thus be less than or equal to this largest possible sum. - The strong inequality follows from the motivation to eliminate the series by way of reducing it to a series for which a formula is known. In the proof so far, the purpose for introducing the inequality in #1 comes from intuition that (the geometric series formula); therefore, if an inequality can be found from that introduces a series with (
*b*−1) in the numerator, and if the denominator term can be further reduced from to , as well as shifting the series indices from 0 to , then both series and (*b*−1) terms will be eliminated, getting closer to a fraction of the form , which is the end-goal of the proof. This motivation is increased here by selecting now from the sum a partial sum. Observe that, for any term in , since*b*≥ 2, then , for all*k*(except for when*n*=1). Therefore, (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 (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 . - For the final inequality , this particular inequality has been chosen (true because
*b*≥ 2, where equality follows if and only if*n*=1) because of the wish to manipulate into something of the form . 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 .

Here the proof will show that the number where c and d are integers and cannot satisfy the inequalities that define a Liouville number. Since every rational number can be represented as such the proof will show that **no Liouville number can be rational**.

More specifically, this proof shows that for any positive integer n large enough that [equivalently, for any positive integer )], no pair of integers exists that simultaneously satisfies the pair of bracketing inequalities

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

Let p and q be any integers with Then,

If then

meaning that such pair of integers would violate the *first* inequality in the definition of a Liouville number, irrespective of any choice of n .

If, on the other hand, since then, since is an integer, we can assert the sharper inequality From this it follows that

Now for any integer the last inequality above implies

Therefore, in the case such pair of integers would violate the *second* inequality in the definition of a Liouville number, for some positive integer n.

Therefore, to conclude, there is no pair of integers with that would qualify such an as a Liouville number.

Hence a Liouville number cannot be rational.

**No Liouville number is algebraic.** The proof of this assertion proceeds by first establishing a property of irrational algebraic numbers. This property essentially says that irrational algebraic numbers cannot be well approximated by rational numbers, where the condition for "well approximated" becomes more stringent for larger denominators. A Liouville number is irrational but does not have this property, so it can't be algebraic and must be transcendental. The following lemma is usually known as **Liouville's theorem (on diophantine approximation)**, there being several results known as Liouville's theorem.

**Lemma:** If is an irrational root of an irreducible polynomial of degree with integer coefficients, then there exists a real number such that for all integers with ,

**Proof of Lemma:** Let be a minimal polynomial with integer coefficients, such that .

By the fundamental theorem of algebra, has at most distinct roots.

Therefore, there exists such that for all we get .

Since is a minimal polynomial of we get , and also is continuous.

Therefore, by the extreme value theorem there exists and such that for all we get .

Both conditions are satisfied for .

Now let be a rational number. Without loss of generality we may assume that . By the mean value theorem, there exists such that

Since and , both sides of that equality are nonzero. In particular and we can rearrange:

**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* and define *m* = *r* + *n*. Since *x* is a Liouville number, there exist integers *a*, *b* with *b* > 1 such that

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

Establishing that a given number is a Liouville number proves that it is transcendental. However, not every transcendental number is a Liouville number. The terms in the continued fraction expansion of every Liouville number are unbounded; using a counting argument, one can then show that there must be uncountably many transcendental numbers which are not Liouville. Using the explicit continued fraction expansion of *e*, one can show that *e* is an example of a transcendental number that is not Liouville. Mahler proved in 1953 that π is another such example.^{[4]}

Consider 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 π.

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

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

From the point of view of measure theory, the set of all Liouville numbers is small. More precisely, its Lebesgue measure, , is zero. The proof given follows some ideas by John C. Oxtoby.^{[5]}^{: 8 }

For positive integers and set:

then

Observe that for each positive integer and , then

Since

and then

Now

and it follows that for each positive integer , has Lebesgue measure zero. Consequently, so has .

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

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

For each positive integer n, set

The set of all Liouville numbers can thus be written as

Each is an open set; as its closure contains all rationals (the from each punctured interval), it is also a dense subset of real line. Since it is the intersection of countably many such open dense sets, L is comeagre, that is to say, it is a *dense* G_{δ} set.

The **Liouville–Roth irrationality measure** (**irrationality exponent,** **approximation exponent,** or **Liouville–Roth constant**) of a real number is a measure of how "closely" it can be approximated by rationals. It is defined by adapting the definition of Liouville numbers: instead of requiring the existence of a sequence of pairs that make the inequality hold for each —a sequence which necessarily contains infinitely many distinct pairs—the irrationality exponent is defined to be the supremum of the set of for which such an infinite sequence exists, that is, the set of such that is satisfied by an infinite number of integer pairs with .^{[6]}^{: 246 } For any value , the infinite set of all rationals satisfying the above inequality yields good approximations of . Conversely, if , then there are at most finitely many with that satisfy the inequality. If is a Liouville number then .

**^**Joseph Liouville (May 1844). "Mémoires et communications".*Comptes rendus de l'Académie des Sciences*(in French).**18**(20, 21): 883–885, 910–911.**^**Baker, Alan (1990).*Transcendental Number Theory*(paperback ed.). Cambridge University Press. p. 1. ISBN 978-0-521-39791-9.**^**Baker 1990, p. 86.**^**Kurt Mahler, "On the approximation of π",*Nederl. Akad. Wetensch. Proc. Ser. A.*, t. 56 (1953), p. 342–366.**^**Oxtoby, John C. (1980).*Measure and Category*. Graduate Texts in Mathematics. Vol. 2 (Second ed.). New York-Berlin: Springer-Verlag. doi:10.1007/978-1-4684-9339-9. ISBN 0-387-90508-1. MR 0584443.**^**Bugeaud, Yann (2012).*Distribution modulo one and Diophantine approximation*. Cambridge Tracts in Mathematics. Vol. 193. Cambridge: Cambridge University Press. doi:10.1017/CBO9781139017732. ISBN 978-0-521-11169-0. MR 2953186. Zbl 1260.11001.

- The Beginning of Transcendental Numbers