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In the 1840s, [[Joseph Liouville]] obtained the first lower bound for the approximation of [[algebraic number]]s: If ''x'' is an irrational algebraic number of degree ''n'' over the rational numbers, then there exists a constant {{nowrap|''c''(''x'') > 0}} such that
:<math> \left| x- \frac{p}{q} \right| > \frac{c(x)}{q^
holds for all integers ''p'' and ''q'' where {{nowrap|''q'' > 0}}.
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holds for every integer {{math|''p''}} and {{math|''q''}} such that {{math|''q'' > 0}}.
In some sense, this result is optimal, as the theorem would be false with ''ε'' = 0. This is an immediate consequence of the upper bounds described below.
=== Simultaneous approximations of algebraic numbers ===
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