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{\huge  \sc   Homework 3} \\
\smallskip
{ Due Date: 04/12/2022}\\
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{Name:  }\\
{\large  }
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\begin{enumerate}





\item[2.1)]
\begin{enumerate}
    
    \item Prove that $\sqrt{3}$ is irrational
    
    \item Prove that there are no rationals $r,s$ such that $\sqrt{3}=r+s\sqrt{2}$.
    
\end{enumerate}

\newpage
\item[2.4]
\begin{enumerate}
\item  Let $a, b$ be rationals and $x$ irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$
\item Let $x, y$ be rationals such that $\frac{x^{2}+x+\sqrt{2}}{y^{2}+y+\sqrt{2}}$ is also rational. Prove that either $x=y$ or $x+y=-1$
\end{enumerate}
\newpage
\item[2.5] Prove that if $n$ is any positive integer, then $\sqrt{n}+\sqrt{2}$ is irrational. 

\newpage 

\item[3.1)] Express the decimal $1.\overline{813}$ as a fraction.
\newpage 
\item[3.3)] Which of the following numbers are rational, and which are irrational? Express those which are rational in the form $\frac{m}{n}$ with $m,n$ integers.
\begin{enumerate}
    \item $0.a_1a_2a_3\ldots$, where for $n=1,2,3\ldots $, the value of $a_n$ is the number $0,1,2,3$ or $4$ which is the remainder on dividing $n$ by 5. 
    
    \item $0.101001000100001000001\ldots$
    
    \item $1.b_1b_2b_3\ldots$, where $b_i=1$ if $i$ is a square, and $b_i=0$ if $i$ is not a square.
    
\end{enumerate}
\newpage 

\item[3.4)] Without using a calculator, find the cube root of 2, correct to 1 decimal place. 
\newpage
\item[3.6)] Show that for an integer $n\geq 2$, the period of $\frac{1}{n}$ is at most $n-1$.
Find the first few values of $n$ for which the period of $\frac{1}{n}$ is equal to $n-1$. Do ou notice anything interesting about the values you've found?
\end{enumerate}



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