What is a surd?
\begin{solutionordottedlines}[2cm]
 A surd is an irrational number that can be expressed in the form of a root, such as \(\sqrt{2}\), which cannot be simplified to a rational number.             
\end{solutionordottedlines}

You must realise that a surd is an \textbf{exact value}, and anything that your calculator spits back at you is only an \emph{approximation}. In your calculator $\sqrt{2} = 1.414213562$, but in truth $\sqrt{2}$ never terminates, it continues forever as it is \emph{irrational}!

Let's round some surds to 3 decimal places:
 \begin{examplebox}                                                                                                                          
     \section{Examples}                                                                                                                     
     \begin{multicols}{4}                                                                                                                    
         \begin{questions}                                                                                                                   
             \Question[1] \(\sqrt{19}=\)\fillin[4.359]{0.7in}                                                                                
             \Question[1] \(\sqrt{37}=\)\fillin[6.083]{0.7in}                                                                                
             \Question[1] \(\sqrt{161}=\)\fillin[12.688]{0.65in}                                                                             
             \Question[1] \(\sqrt{732}=\)\fillin[27.055]{0.65in}                                                                             
         \end{questions}                                                                                                                     
     \end{multicols}                                                                                                                         
 \end{examplebox}                                                                                                                            

Now we shall find the \emph{exact} lengths of the following sides. Realise that unlike before our side lengths are not whole or even \emph{rational} numbers.

 \begin{exercisebox}                                                                                                                         
     \section{Exercises}                                                                                                                    
         \begin{questions}                                                                                                                   
             \begin{multicols}{3}                                                                                                            
                 \Question[1]\,                                                                                                              
                 \drawTriangle{bottom left}{5.83}{3}{5}{$3$}{$5$}{$x$}{0.5}                                                                  
                 \begin{solutionordottedlines}[1in]                                                                                          
                 \( x = \sqrt{5.83^2 - 3^2} = \sqrt{34} \approx 5.831 \)                                                                     
                 \end{solutionordottedlines}                                                                                                 
                 \Question[1]\,                                                                                                              
                 \drawTriangle{top left}{8.944}{8}{4}{$4$}{$8$}{$y$}{0.5}                                                                    
                 \begin{solutionordottedlines}[1in]                                                                                          
                 \( y = \sqrt{8^2 - 4^2} = \sqrt{48} \approx 6.928 \)                                                                        
                 \end{solutionordottedlines}                                                                                                 
                 \Question[1]\,                                                                                                              
                 \drawTriangle{bottom right}{7}{2}{6.708}{$2$}{$x$}{$7$}{0.5}                                                                
                 \begin{solutionordottedlines}[1in]                                                                                          
                 \( x = \sqrt{7^2 - 2^2} = \sqrt{45} \approx 6.708 \)                                                                        
                 \end{solutionordottedlines}                                                                                                 
             \end{multicols}                                                                                                                 
         \Question[2] A door frame has height $1.8$m and width $1$m. Will a square piece of board $2.1$m wide fit through the opening?       
             \begin{solutionordottedlines}[1.5in]                                                                                            
             The diagonal of the door frame is \(\sqrt{1.8^2 + 1^2} = \sqrt{4.24} \approx 2.059\), which is greater than $2.1$m, so the boar 
 will fit.                                                                                                                                   
             \end{solutionordottedlines}                                                                                                     
         \Question[3] A signwriter leans his ladder against a wall so that he can paint a sign. The wall is vertical and the ground in front 
 of the wall is horizontal. The signwriter's ladder is $4$m long.\\If the signwriter wants the top of the ladder to be $3.8$m above the grou 
 when leaning against the wall, how far, correct to $1$ decimal place should the foot of the ladder be placed from the wall?                 
             \begin{solutionordottedlines}[1.5in]                                                                                            
             The distance from the wall is \(\sqrt{4^2 - 3.8^2} = \sqrt{0.36} \approx 0.6\)m.                                                
             \end{solutionordottedlines}                                                                                                     
         \end{questions}                                                                                                                     
 \end{exercisebox}