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Liouville also exhibited examples of Liouville numbers [1] thereby establishing the existence of transcendental numbers for the first time. [2] One of these examples is Liouville's constant L = 0.110001000000000000000001 … , {\displaystyle L=0.110001000000000000000001\ldots ,}
This might seem to be a much stronger result than Liouville's theorem, but it is actually an easy corollary. If the image of f {\displaystyle f} is not dense, then there is a complex number w {\displaystyle w} and a real number r > 0 {\displaystyle r>0} such that the open disk centered at w {\displaystyle w} with radius r {\displaystyle r} has ...
The Liouville lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if n is the product of an even number of prime numbers , and −1 if it is the product of an odd number of primes.
ω(x, 1) is often called the measure of irrationality of a real number x. For rational numbers, ω(x, 1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality. Roth's theorem says that irrational real algebraic numbers have measure of irrationality 1.
In Hamiltonian mechanics, see Liouville's theorem (Hamiltonian) and Liouville–Arnold theorem; In linear differential equations, see Liouville's formula; In transcendence theory and diophantine approximations, the theorem that any Liouville number is transcendental; In differential algebra, see Liouville's theorem (differential algebra)
This knowledge enabled Liouville, in 1844, to produce the first explicit transcendental number. Later, the proofs that π and e are transcendental were obtained by a similar method. Diophantine approximations and transcendental number theory are very close areas that share many theorems and methods.
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