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Measure of irrationality

From Encyclopedia of Mathematics
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of a real number

The function

where the minimum is over all pairs of integral rational numbers such that

The concept of the measure of irrationality is a particular case of those of the measure of linear independence and the measure of transcendency (cf. Linear independence, measure of; Transcendency, measure of). The measure of irrationality indicates how "well" the number can be approximated by rational numbers. For all real irrational numbers one has

but for any and almost-all (in the sense of the Lebesgue measure) real numbers ,

where . However, for any function with as and , there exists a number such that for all ,

References

[1] A.Ya. Khinchin, "Continued fractions" , Univ. Chicago Press (1964) (Translated from Russian)
How to Cite This Entry:
Measure of irrationality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measure_of_irrationality&oldid=32680
This article was adapted from an original article by A.I. Galochkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article