Rational Approximation
If 
is any number and 
and 
are integers,
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)
|
(Hilbert and Cohn-Vossen 1999, pp. 40-44).
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 
.
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Freiman's constant
