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:<math> \left | \alpha -\frac{p}{q} \right | < \frac{1}{q^2} </math>
is satisfied by infinitely many integers ''p'' and ''q''. This shows that any irrational number has [[irrationality measure]] at least 2. This corollary also shows that the [[Thue–Siegel–Roth theorem]], a result in the other direction, provides essentially the tightest possible bound, in the sense that the bound on rational approximation of [[algebraic number]]s cannot be improved by increasing the exponent beyond 2. The Thue–Siegel–Roth theorem uses advanced techniques of number theory, but many simpler numbers such as the [[golden ratio]] <math>(1+\sqrt{5})/2</math> can be much more easily verified to be inapproximable beyond exponent 2. This exponent is referred to as the [[irrationality measure]].
==Simultaneous version==
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