Liouville Number

A Liouville number is a Transcendental Number which has very close Rational Number approximations. An Irrational Number $\beta$ is a Liouville number if, for any $n$, there exist an infinite number of pairs of Integers $p$ and $q$ such that

$$0<\left\vert{\beta-{p\over q}}\right\vert<{1\over q^n}.$$

Mahler (1953) proved that $\pi$ is not a Liouville number.

See also Liouville's Constant, Liouville's Rational Approximation Theorem, Roth's Theorem, Transcendental Number

References

Mahler, K. ``On the Approximation of $\pi$.'' Nederl. Akad. Wetensch. Proc. Ser. A. 56/Indagationes Math. 15, 30-42, 1953.