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Simple Continued Fraction


A simple continued fraction is a special case of a generalized continued fraction for which the partial numerators are equal to unity, i.e., a_n=1 for all n=1, 2, .... A simple continued fraction is therefore an expression of the form

 b_0+1/(b_1+1/(b_2+1/(b_3+...))).
(1)

When used without qualification, the term "continued fraction" is often used to mean "simple continued fraction" or, more specifically, regular (i.e., a simple continued fraction whose partial denominators b_0, b_1, ... are positive integer; Rockett and Szüsz 1992, p. 3). Care must therefore be taken to identify the intended meaning based on the context in which such terminology is encountered.

A simple continued fraction can be written in a compact abbreviated notation as

 x=K_(k=1)^N1/(b_k)
(2)

or

 x=[b_0;b_1,b_2,b_3,...],
(3)

where N may be finite (for a finite continued fraction) or infty (for an infinite continued fraction). In contexts where only simple continued fractions are considered, the partial denominators are often denoted [a_0;a_1,a_2,...] instead of [b_0;b_1,b_2,...] (e.g., Rockett and Szüsz 1992, p. 3), a practice which unfortunately conflicts with the common notation for generalized continued fractions in which a_n denotes a partial numerator.

Further care is needed when encountering bracket notation for simple continued fractions since some authors replace the semicolon with a normal comma and begin indexing the terms at b_1 instead of b_0, writing [b_0,b_1,b_2,...] instead of [b_0;b_1,b_2,...] or b_0+[b_1,b_2,...], causing ambiguity in the meaning of the initial term and resulting in the parity of certain fundamental results in continued fraction theory to be reversed. To complicate matters a bit further, Gaussian brackets use the notation [a_1,a_2,...,a_n] to denote a different (but closely related) combination of partial denominators.

The terms b_0 through b_(n-1) of the simple continued fraction of a number x can be computed in the Wolfram Language using the command ContinuedFraction[x, n]. Similarly, the n convergent of simple continued fraction with partial denominators b_k can be continued using ContinuedFractionK[a[k], {k, n}], where n may be Infinity.


See also

Continued Fraction, Convergent, Gaussian Brackets, Partial Denominator, Partial Numerator, Regular Continued Fraction

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References

Rockett, A. M. and Szüsz, P. Continued Fractions. New York: World Scientific, 1992.

Referenced on Wolfram|Alpha

Simple Continued Fraction

Cite this as:

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

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