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Concept
Liouville number
Summary
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
Liouville numbers are "almost rational", and can thus be approximated "quite closely" by sequences of rational numbers. Precisely, these are transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number can be. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.
It is known that pi and e are not Liouville numbers.
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:
L = 0.11000100000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001...
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 ak ∈ {0, 1, 2, ..., b−1} for all k, so at most ak = b−1. The largest possible sum would occur if the sequence of integers (a1, a2, ...) were (b−1, b−1, ...), i.e. ak = 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.
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