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{{main|Thue–Siegel–Roth theorem}}
Over more than a century, there were many efforts to improve Liouville's theorem: every improvement of the bound enables us to prove that more numbers are transcendental. The main improvements are due to {{harvs|first=Axel|last=Thue|authorlink=Axel Thue|year=1909|txt}}, {{harvs|frst=Carl Ludwig|last=Siegel|authorlink=Carl Ludwig Siegel|year=1921|txt}}, {{harvs|first=Freeman|last=Dyson|authorlink=Freeman Dyson|year=1947|txt}}, and {{harvs|first=Klaus|last=Roth|authorlink=Klaus Roth|year=1955|txt}}, leading finally to the Thue–Siegel–Roth theorem: If {{math|''x''}} is an irrational algebraic number and {{math|''ε > 0''}}
:<math>
\left| x- \frac{p}{q} \right|>\frac{c(x, \varepsilon)}{q^{2+\varepsilon}}
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