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Continued Fraction

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The term "continued fraction" is used to refer to a class of expressions of which generalized continued fraction of the form

b_0+(a_1)/(b_1+(a_2)/(b_2+(a_3)/(b_3+...))) =b_0+K_(n=1)^infty(a_n)/(b_n)

(and the terms may be integers, reals, complexes, or functions of these) are the most general variety (Rocket and Szüsz 1992, p. 1).

Wallis first used the term "continued fraction" in his Arithmetica infinitorum of 1653 (Havil 2003, p. 93), although other sources list the publication date as 1655 or 1656. An archaic word for a continued fraction is anthyphairetic ratio.

The simple continued fraction takes a_n=1 for all n, leaving

b_0+1/(b_1+1/(b_2+1/(b_3+...)))=b_0+K_(n=1)^infty1/(b_n).

If b_0 is an integer and the remainder of the partial denominators b_k for k>0 are positive integers, the continued fraction is known as a regular continued fraction.

The most successful algorithm employed by the Ramanujan Project (Raayoni et al. 2021) relied on a brute-force search over the space of polynomial continued fractions to find new formulas for mathematical constants. Elimelech et al. (2023) subsequently used algorithm involving factorial reduction to search for new polynomial continued fraction formulas, discovering hundreds of new formulas for mathematical constants, including e, ln2, pi, and zeta(3).

See also

Continued Fraction Constants, Convergent, Generalized Continued Fraction, Regular Continued Fraction, Simple Continued Fraction Explore this topic in the MathWorld classroom

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References

Ben David, N.; Nimri, G.; Mendlovic, U.; Manor, Y.; and Kaminer, I. "On the Connection Between Irrationality Measures and Polynomial Continued Fractions." 5 Nov 2021. https://arxiv.org/abs/2111.04468.Cuyt, A. A.; Petersen, V.; Verdonk, B.; Waadeland, H.; and Jones, W. B. Handbook of Continued Fractions for Special Functions. Dordrecht, Netherlands: Springer, 2008.Elimelech, R.; David, O.; De la Cruz Mengual, C.; Kalisch, R.; Berndt, W.; Shalyt, M.; Silberstein, M.; Hadad, Y.; and Kaminer, I. "Algorithm-Assisted Discovery of an Intrinsic Order Among Mathematical Constants." 22 Aug 2023. https://arxiv.org/abs/2308.11829.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Raayoni, G; Gottlieb, S.; Manor, Y.; Pisha, G.; Harris, Y.; Mendlovic, U.; Haviv, D.; Hadad, Y.; and Kaminer, I. "Generating Conjectures on Fundamental Constants With the Ramanujan Machine." Nature 590, 67-73, 2021.Rockett, A. M. and Szüsz, P. Continued Fractions. New York: World Scientific, 1992.Wallis, J. Arithmetica Infinitorum. Oxford, England: Typis Leon. Lichfield, Impensis Tho. Robinson, 1656.

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Continued Fraction

Cite this as:

Weisstein, Eric W. "Continued Fraction." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ContinuedFraction.html

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