: Mathématiques: Liouville number

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Liouville number

In number theory, a Liouville number is a real number ${\\displaystyle x}$ with the property that, for every positive integer ${\\displaystyle n}$ , there exists a pair of integers ${\\displaystyle (p,q)}$ with ${\\displaystyle q>1}$ such that ${\\displaystyle 0<\\left|x-{\ rac {p}{q}}\ ight|<{\ rac {1}{q^{n}}}.}$ 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 $π$ and $e$ are not Liouville numbers.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Liouville_number)

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