Measure of irrationality

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2020 Mathematics Subject Classification: Primary: 11J82 [MSN][ZBL]

of a real number $\xi$

The function


where the minimum is over all pairs $h_0,h_1$ of integral rational numbers such that

$$|h_0|,|h_1|\leq H,\quad |h_0|+|h_1|\neq0.$$

The concept of the measure of irrationality is a particular case of those of the measure of linear independence and the measure of transcendency. The measure of irrationality indicates how "well" the number $\xi$ can be approximated by rational numbers. For all real irrational numbers one has


but for any $\epsilon>0$ and almost-all (in the sense of the Lebesgue measure) real numbers $\xi$,


where $C=C(\epsilon,\xi)>0$. However, for any function $\phi$ with $\phi(H)\to0$ as $H\to\infty$ and $\phi(H)>0$, there exists a number $\xi_\phi$ such that for all $H\geq1$,



[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:
This article was adapted from an original article by A.I. Galochkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article