As Lagrange showed, any irrational number has an infinity of rational approximations which satisfy
(1)
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Furthermore, if there are no integers with and (corresponding to values of associated with the golden ratio through their continued fractions), then
(2)
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and if values of associated with the silver ratio are also excluded, then
(3)
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In general, even tighter bounds of the form
(4)
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can be obtained for the best rational approximation possible for an arbitrary irrational number , where the are called Lagrange numbers and get steadily larger for each "bad" set of irrational numbers which is excluded.