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In number theory, a Liouville number is an irrational number ''x'' with the property that, for every 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. == The existence of Liouville numbers (Liouville's constant) == Here we show that Liouville numbers exist by exhibiting a construction that produces such numbers. For any integer ''b'' ≥ 2, and any sequence of integers (''a''1, ''a''2, …, ), such that ''a''''k'' ∈ , ∀''k'' ∈ , define the number : ...where the last equality follows from the fact that : Therefore, we conclude that any such ''x'' is a Liouville number. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Liouville number」の詳細全文を読む スポンサード リンク
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