In number theory, a Liouville number is a real number x with the property that, for any positive integer n, there exist integers p and q with q > 1 and such that
A Liouville number can thus be approximated "quite closely" by a sequence of rational numbers. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.
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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 irrational. Assume otherwise; then there exist integers c, d with d > 0 and 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 p_{n} and q_{n} as follows:
then we have for all positive integers n
Uncountability
Consider, for example, the number
3.14(3 zeros)1(17 zeros)5(95 zeros)9(599 zeros)2...
where the digits are zero except in positions 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 nonterminating decimal with its nonzero digits similarly situated, satisfies the definition of Liouville number. Since the set of all sequences of nonnull digits has the cardinality of the continuum, the same thing occurs with the set of all Liouville numbers. Moreover, the Liouville numbers form a 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 L is small. More precisely, its Lebesgue measure is zero. The proof given follows some ideas by John C. Oxtoby.^{[1]}^{:8}
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