Measure of irrationality
From Encyclopedia of Mathematics
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=11646
Measure of irrationality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measure_of_irrationality&oldid=11646
This article was adapted from an original article by A.I. Galochkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article