Liouville number [SEP] named after
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", "a", …, ), such that "a" ∈ {0, 1, 2, …
, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational numbers than can any irrational algebraic number, and this class of numbers are called Liouville numbers, named in honour of him. Liouville showed that all Liouville numbers are transcendental.
The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was , by Charles Hermite in 1873.
In 1874, Georg Cantor proved that the algebraic numbers are countable and