The basis of this chapter is the Euclidean algorithm, which has been part of mathematics since at least the fourth century BC, predating Archimedes. From the Euclidean algorithm we immediately obtain the expression of any rational number in the form of a finite continued fraction. A study of continued fractions shows us that they provide a more natural method of expressing any real number in terms of integers than the usual decimal expansion. An investigation of the “worst” case in applying the Euclidean algorithm leads to the Fibonacci sequence and so to other sequences generated by a linear recurrence relation.
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