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 
.
REFERENCES:
Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 41, 1999.
