If
is any number and
is any Integer, then there is a Rational Number
for which
![\begin{displaymath}
0\leq r-{m\over n} < {1\over n}.
\end{displaymath}](r_700.gif) |
(1) |
If
is Irrational and
is any Whole Number, there is a Fraction
with
and for which
![\begin{displaymath}
0\leq r-{m\over n} < {1\over nk}.
\end{displaymath}](r_702.gif) |
(2) |
Furthermore, there are an infinite number of Fractions
for which
![\begin{displaymath}
0\leq r-{m\over n} < {1\over n^2}.
\end{displaymath}](r_703.gif) |
(3) |
Hurwitz has shown that for an Irrational Number
![\begin{displaymath}
\left\vert{\zeta-{h\over k}}\right\vert < {1\over ck^2},
\end{displaymath}](r_705.gif) |
(4) |
there are infinitely Rational Numbers
if
, but if
, there
are some
for which this approximation holds for only finitely many
.
© 1996-9 Eric W. Weisstein
1999-05-25