This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations : in fact an immediate consequence is that for a given irrational ?, the inequality
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Siegel's result was ineffective ( see effective results in number theory ), since Thue's method in diophantine approximation also is ineffective in describing possible very good rational approximations to algebraic numbers.
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They are intimately connected with the golden ratio; for example, the closest rational approximations to the ratio are 2 / 1, 3 / 2, 5 / 3, 8 / 5, . . ..
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One can choose to define a " best rational approximation " to a real number as a rational number | } },, that is closer to than any approximation with a smaller or equal denominator.
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In mathematics, the "'Pell numbers "'are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2.
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Thus when we introduce irrational number in an error-free computation having a finite number of arithmetic operations, we should use the absolute best digit rational approximation and not the other two approximations for committing minimal error.
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Since it is a convergent of the continued fraction representation of the square root of two, any better rational approximation has a denominator not less than 169, since is the next convergent with an error of approx ..
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The concrete information that was left theoretically incomplete included lower bounds for class numbers ( ideal class groups for some families of number fields grow ); and bounds for the best rational approximations to algebraic numbers in terms of denominators.
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The relative performance of software BKM implementation in comparison to other methods such as polynomial or rational approximations will depend on the availability of fast multi-bit shifts ( i . e . a barrel shifter ) or hardware floating point arithmetic.
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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 limits on rational approximation of algebraic numbers cannot be improved by lowering the exponent 2 + & epsilon; beyond 2.
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