continued fraction

English

A regular continued fraction (in which every numerator is 1)

Alternative forms

• continued-fraction (attributive use)

Noun

continued fraction (plural continued fractions)

1. (mathematics, number theory) 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.
   • 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:

     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.

   • 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.

   • 2009, M. Welleda Baldoni, Ciro Ciliberto, G.M. Piacentini Cattaneo, translated by Daniele Gewurz, Elementary Number Theory, Cryptography and Codes, page 48:

     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.

Usage notes

The initial integer may be zero or even negative; subsequent non-numerator terms should be positive (while not strictly necessary, it is easier to prove convergence with positive terms).
Generally, it is assumed that every numerator is 1; if distinction is necessary, such a fraction may be called regular or simple or be said to be in canonical form. (See continued fraction on Wikipedia.Wikipedia )

Derived terms

• Euler's continued fraction formula
• general continued fraction
• generalized continued fraction
• regular continued fraction
• simple continued fraction
• periodic continued fraction
• finite continued fraction
• infinite continued fraction
• Rogers-Ramanujan continued fraction

Translations

number theory

• Finnish: ketjumurtoluku
• French: fraction continue f
• German: Kettenbruch (de) m
• Icelandic: keðjubrot n
• Italian: frazione continua f
• Polish: ułamek łańcuchowy m
• Swedish: kedjebråk (sv) n

See also

• convergent