Editing Diophantine approximation (section)

=== Approximation of algebraic numbers, Liouville's result ===
{{main|Liouville number}}

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^n}</math>

holds for all integers ''p'' and ''q'' where {{nowrap|''q'' > 0}}.

This result allowed him to produce the first proven example of a transcendental number, the [[Liouville constant]]
:<math>
\sum_{j=1}^\infty 10^{-j!} = 0.110001000000000000000001000\ldots\,,
</math>
which does not satisfy Liouville's theorem, whichever degree ''n'' is chosen.

This link between Diophantine approximations and transcendental number theory continues to the present day. Many of the proof techniques are shared between the two areas.