Joseph Liouville and his Numbers (especially his constant)

Number theory. Whoo!

In number theory, let there be a number x where, for any positive integer n, there are infinitely many number pairs (p, q) where q > 1 that:

0 < |x – p/q| < 1/q^n

x is the Liouville number.

Liouville numbers can be described as “almost rational”, and can be approximated by sequences of rational numbers. In other words, Liouville numbers are transcendentals that can be more closely approximated than any other algebraic irrational.

In 1844, Joseph Liouville proved that all Liouville numbers were transcendental. Maybe, if I can find the proof, I might post why, but not now.

One Liouville number is the Liouville Constant, which takes the form of:

 L=sum_(n=1)^infty10^(-n!)=0.110001000000000000000001...
0.1100010000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001… I’m not sure if that’s 120 decimal digits.

One thing: Liouville’s Constant (call it L; TapL hahaha) almost satisfies 10x^6 -75x^3 – 190x + 21 = 0, if the solution was rounded, then maybe.

If my mind comes to it, I might go into more detail

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