FWIW, all of the convergents from the continued fraction expansion of a number have "quality" greater than 2. Conversely, the Thue–Siegel–Roth theorem says that any algebraic number and any epsilon > 0, the algebraic number has only finitely many approximations with quality more than 2 + epsilon; so the "quality" values of successive best approximations are a series which converges to 2 from above.
This was how the first known transcendental numbers -- the Liouville numbers -- were constructed: They have approximations with unbounded quality, thus they cannot be algebraic. In practice, however, this isn't a very useful method: "Almost all" transcendental numbers also obey the Thue–Siegel–Roth theorem.
This was how the first known transcendental numbers -- the Liouville numbers -- were constructed: They have approximations with unbounded quality, thus they cannot be algebraic. In practice, however, this isn't a very useful method: "Almost all" transcendental numbers also obey the Thue–Siegel–Roth theorem.