\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} A rational number is a quotient of
two integers. Thus, ${27}\over{894}$ is a rational number, and so is
${{17}\over{44}}+{{90}\over{2,103}}-{{1}\over{337}}$
(``simplification'' is {\it not}\/ necessary in this problem -- sums,
products, etc.~of rational numbers {\it are}\/ rational numbers).

\medskip

\noindent a) Find a rational number that is within $10^{-100}$ of
$\sin(.4)$. 

\medskip

\noindent b) Find a rational number that is within $10^{-100}$ of
$\sin(1.4)$. 

\medskip

\noindent c) Find a rational number that is within $10^{-100}$ of
$e^{2.8}$.

\smallskip

\noindent {\bf Comment} You are {\it not} asked to exhibit, for
example, the decimal approximation implied. You are only asked to show
the rational numbers requested and give reasons why your
approximations are correct. There are many possible correct answers to
this problem.










\vfil\eject\end