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Nearest Integer Continued Fraction


Every irrational number x can be expanded in a unique continued fraction expansion

 x=b_0+(e_1)/(b_1+(e_2)/(b_2+(e_3)/(b_3+...)))=[b_0;e_1b_1,e_2b_2,e_3b_3,...]

such that b_0 in Z, x-b_0 in [-1/2,1/2), e_n=+/-1, b_n in N and e_(n+1)+b_n>=2 for n>=1. This continued fraction expansion is known as the nearest integer continued fraction expansion of x.

For example, the nearest integer continued fraction expansion of e is given by

 [3;-4,-2,2k+5^_]_(k=0)^infty.

See also

Continued Fraction

This entry contributed by David Terr

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References

Hartano, Y. and Kraaikamp, C. "A Note on Hurwitzian Numbers." http://ssor.twi.tudelft.nl/~cork/hurwitz.pdf.

Referenced on Wolfram|Alpha

Nearest Integer Continued Fraction

Cite this as:

Terr, David. "Nearest Integer Continued Fraction." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/NearestIntegerContinuedFraction.html

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