A rational number $Q$ may be represented by two integers $N$ the numerator and $D$ the denominator.

\begin{align}
	Q &= \frac{N}{D} 
\end{align}			

Compared to floating point arithmetic which is generally inexact, rational arithmetic including the division operation is always exactly representable as another rational number. However, a \emph{fixed size} rational representation is of rather limited use as $D$ will always grow after repeated operations and overflow. Use of arbitrary sized integers as described in section \ref{Big Integers} and implemented by GMP\cite{granlund2004GMP} overcomes this issue; however as we will see in Chapter \ref{Results and Discussion} there can be a significant performance cost associated with Rationals.
  
	\begin{align}
		N = \sum_{i=0}^{S} n_i \beta^{i} & \text{ and }
		D = \sum_{i=0}^{S} d_i \beta^{i} \text{ where $S$ grows as needed}
	\end{align}	
	
