Generalized Continued Fraction - History of Continued Fractions

History of Continued Fractions

The story of continued fractions begins with the Euclidean algorithm, a procedure for finding the greatest common divisor of two natural numbers m and n. That algorithm introduced the idea of dividing to extract a new remainder – and then dividing by the new remainder again, and again, and again.

Nearly two thousand years passed before Rafael Bombelli devised a technique for approximating the roots of quadratic equations with continued fractions. Now the pace of development quickened. Just 24 years later Pietro Cataldi introduced the first formal notation for the generalized continued fraction. Cataldi represented a continued fraction as

& & &

with the dots indicating where the next fraction goes, and each & representing a modern plus sign.

Late in the seventeenth century John Wallis introduced the term "continued fraction" into the mathematical literature. New techniques for mathematical analysis (Newton's and Leibniz's calculus) had recently exploded onto the scene, and a generation of Wallis' contemporaries put the new word to use right away.

In 1748 Euler published a very important theorem showing that a particular kind of continued fraction is equivalent to a certain very general infinite series. Euler's continued fraction theorem is still of central importance in modern attempts to whittle away at the convergence problem.

Continued fractions can also be applied to problems in number theory, and are especially useful in the study of Diophantine equations. In the late eighteenth century Lagrange used continued fractions to construct the general solution of Pell's equation, thus answering a question that had fascinated mathematicians for more than a thousand years. Amazingly, Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic and that, if the period is of length p > 1, it contains a palindromic string of length p - 1.

In 1813 Gauss used a very clever trick with the complex-valued hypergeometric function to derive a versatile continued fraction expression that has since been named in his honor. That formula can be used to express many elementary functions (and even some more advanced functions, like the Bessel functions) as rapidly convergent continued fractions valid almost everywhere in the complex plane.