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The irrationality measure should be specified - it's the irrationality exponent. Link now points to that specific section, irrationality measure #irrationality_exponent Tags: Mobile edit Mobile web edit |
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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#irrationality_exponent|irrationality exponent]] at least 2.
The [[Thue–Siegel–Roth theorem]] says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 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.
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