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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due May 18 (Friday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 453 $|$ Real Analysis I $|$ Summer
    I 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Francis Su's essay on good mathematical writing (posted
on the course website). \\


\begin{enumerate}

\item Answer the following questions from the directly from the essay
  on good mathematical writing.
  \begin{enumerate}
  \item What is a good rule of thumb for what you should assume of your
    audience as you write your homework sets?  
  \item Is chalkboard writing formal or informal writing?  
  \item Why is the proof by contradiction on page 3 not really a
    proof by contradiction?
  \item Name three things a lazy writer would do that a good writer
    would not.  
  \item What's the difference in meaning between these
    three phrases? 
    \[
    \mbox{``Let $A=12$.''}\qquad\qquad\mbox{``So $A=12$.''}
    \qquad\qquad\mbox{``$A=12$.''}
    \]
  \end{enumerate}    


\item Let $X=\Z\setminus\Z_{\neq 0}=\{(m,n)\mid m,n\in\Z,\,n\neq
  0\}$. That is, $X$ consists of all \emph{ordered pairs} $(m,n)$ of
  integers with $n\neq 0$. Consider an \emph{equivalence relation}
  $\sim$ on $X$ defined as
  \[
  a/b\,\sim c/d \quad\mbox{if}\quad ad=bc.
  \]
  The set $\Q:=X/\!\sim$ of equivalence classes are called the
  \emph{rational numbers}. We typically denote the class $[m/n]$ by
  any representative, e.g., $m/n$.
  \begin{enumerate}
  \item Define \emph{addition} of rational numbers by
    \[
    a/b+c/d=[ad+bc]/[bd].
    \]
    Show that this is \emph{well-defined}, which means that it does
    \emph{not} depend on the choice of representative. Specifically,
    show that if $w/x\sim a/b$ and $y/z\sim c/d$, then
    $w/x+y/z=a/b+c/d$.
  \item Define a \emph{multiplication} of rational numbers, and show that it
    is well-defined. \\
  \end{enumerate}


\item If $r$ is rational $(r\neq 0)$ and $x$ is irrational, prove that
  $r+x$ and $rx$ are irrational. Do not make any references to
  Dedekind cuts -- assume that $x$ being irrational means only that it
  cannot be written as a fraction of two integers.  \\


\item Prove that there is no rational number whose square is $20$.

\end{enumerate}

\end{document}



