The Fundamental Correspondence

The method that Bernstein, Levesque, and Rhin used to verify that a certain set of partial quotients were produce by their families relied on mathematical induction, but it was algebraically complicated. The method used to prove the theorems in Section , which is largely used and developed by Madden, makes the process of verifying that a certain set of partial quotients comes from a given family less complicated and less time consuming. The method accomplishes this by taking advantage of the fundamental correspondence between continued fractions and matrices, which is stated in the following theorem.

Theorem 4   If x is a real number whose continued fraction expansion is $x = [q_0, q_1, \ldots, q_n]$, then

$$\left[ \begin{matrix} 0 & 1\\ 1 & q_0\\ \end{matrix} \ri... ...ix} Q_{n-1} & Q_n\\ P_{n-1} & P_n\\ \end{matrix} \right],$$

where P_k/Q_k is the kth convergent of x.

For the remainder of the report, let $\{q_0, q_1, \ldots, q_n\}$denote the matrix product

$$\left[ \begin{matrix} 0 & 1\\ 1 & q_0\\ \end{matrix} \ri... ...t[ \begin{matrix} 0 & 1\\ 1 & q_n\\ \end{matrix} \right].$$

At first, this correspondence between continued fractions and matrices might not seem to be very useful, but when it is used in conjunction with fractional linear transformations, it becomes a powerful tool for the continued fractions of surds. Let $\{q_0, q_1, \ldots, q_n\}(x)$ denote the fractional linear transformation associated with $\{q_0, q_1, \ldots, q_n\}$, where

$$\left[ \begin{matrix} a & b\\ c & d\\ \end{matrix} \right](x) = \frac{ax + b}{cx + d}$$

In general, the relationship between continued fractions and fractional linear transformations is the following.

Theorem 5   $\beta = [q_0, ..., q_n, \alpha]$ if and only if $\{q_0, \ldots, q_n\}(\frac{1}{\alpha}) = \frac{1}{\beta}$.

Now that this theorem has been established, it is possible to prove a proposition that is of prime importance in the the study of the continued fractions of pure quadratic irrationals.