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
,