Liouville numbers are a special class of real numbers that can be approximated by rational numbers with extraordinary accuracy. Specifically, a number is called a Liouville number if for every positive integer n, there exist infinitely many pairs of integers (p, q) such that $$|x - \frac{p}{q}| < \frac{1}{q^{n}}$$. This concept is pivotal in understanding Diophantine approximation and provides crucial insights into transcendence theory, as Liouville numbers are examples of transcendental numbers, which are not roots of any non-zero polynomial equation with integer coefficients.
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