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Rational Approximation

If $r$ is any number and $n$ is any Integer, then there is a Rational Number $m/n$ for which

0\leq r-{m\over n} < {1\over n}.
\end{displaymath} (1)

If $r$ is Irrational and $k$ is any Whole Number, there is a Fraction $m/n$ with $n\leq k$ and for which
0\leq r-{m\over n} < {1\over nk}.
\end{displaymath} (2)

Furthermore, there are an infinite number of Fractions $m/n$ for which
0\leq r-{m\over n} < {1\over n^2}.
\end{displaymath} (3)

Hurwitz has shown that for an Irrational Number $\zeta$
\left\vert{\zeta-{h\over k}}\right\vert < {1\over ck^2},
\end{displaymath} (4)

there are infinitely Rational Numbers $h/k$ if $0<c\leq \sqrt{5}$, but if $c>\sqrt{5}$, there are some $\zeta$ for which this approximation holds for only finitely many $h/k$.

© 1996-9 Eric W. Weisstein