If is any number and is any Integer, then there is a Rational Number for which

(1) 
If is Irrational and is any Whole Number, there is a Fraction with
and for which

(2) 
Furthermore, there are an infinite number of Fractions for which

(3) 
Hurwitz has shown that for an Irrational Number

(4) 
there are infinitely Rational Numbers if
, but if , there
are some for which this approximation holds for only finitely many .
© 19969 Eric W. Weisstein
19990525