Recall from Chapter 16 that the sequence of ratios of consecutive Fibonacci numbers approaches the golden ratio as . Interestingly, we can employ these ratios, coupled with Fibonacci recurrence, to generate fractional numbers of a very special nature, called continued fractions1 . The English mathematician John Wallis (1616–1703) coined the term continued fractions. Some continued fractions have finite decimal expansions, while others do not. We will now begin our pursuit with some basic vocabulary and a few characterizations of continued fractions.
19.1 FINITE CONTINUED FRACTIONS
A finite continued fraction is a multi-layered fraction of the form
where each is a real number; ; and . The numbers are the partial quotients of the finite continued fraction. ...