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Approximation by Rationals

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This Demonstration illustrates that any real number can be approximated by a rational number to within an error equal to half its denominator. In symbols, for any real and natural number , there is a natural number such that .

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This can be seen by partitioning the real line into the half-open intervals of length , . The number must fall into such an interval.

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Contributed by: Izidor Hafner (March 2011)
Open content licensed under CC BY-NC-SA

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