continued fraction meaning: definition, quotations

continued fraction

noun

Mathematics, Number theory, Sciences A compound numerical expression consisting of an integer plus a fraction whose numerator is a positive integer and whose denominator is a continued fraction (an integer plus a fraction), and so on, with finite or infinite recursion.

Several results focus on the famous Rogers–Ramanujan continued fraction [47], [48, pp. 214-215], the only continued fraction appearing in Ramanujan's published papers.1992, G. E. Andrews, B. C. Berndt, L. Jacobsen, R. L. Lamphere, “The Continued Fractions Found in the Unorganized Portions of Ramanujan's Notebooks”, in Memoirs of the American Mathematical Society, volume 99, number 477, page 1:
2000, Andrew Zardecki, 18: Continued Fractions in Time Series Forec[a]sting, Da Ruan (editor), Fuzzy Systems and Soft Computing in Nuclear Engineering, Physica-Verlag, Studies in Fuzziness and Soft Computing, page 397, We achieve this by using well-known examples from the number theory pertaining to the continued fractions. Any sequence of natural numbers drawn from the probability distribution of the quotients of the continued fraction corresponding to an irrational number represents a typical sequence, in the sense that almost all sequences of quotients have this distribution.

1 more example

We have seen that all rational numbers, and no other number, can be expressed as finite simple continued fractions. The main reason of interest of continued fractions, however, is in their application to the representation of irrational numbers. To that end we shall need infinite simple continued fractions.2009, M. Welleda Baldoni, Ciro Ciliberto, G.M. Piacentini Cattaneo, translated by Daniele Gewurz, Elementary Number Theory, Cryptography and Codes, page 48:

Derived terms: Euler's continued fraction formula, finite continued fraction, general continued fraction, generalized continued fraction, infinite continued fraction, periodic continued fraction, regular continued fraction, Rogers-Ramanujan continued fraction, simple continued fraction
Related terms: convergent