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ContinuedFraction[x,n]

generates a list of the first n terms in the continued fraction representation of x.

ContinuedFraction[x]

generates a list of all terms that can be obtained given the precision of x.

Details
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Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Applications  
Properties & Relations  
Neat Examples  
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ContinuedFraction

ContinuedFraction[x,n]

generates a list of the first n terms in the continued fraction representation of x.

ContinuedFraction[x]

generates a list of all terms that can be obtained given the precision of x.

Details

  • The continued fraction representation {a1,a2,a3,…} corresponds to the expression a1+1/(a2+1/(a3+…)).
  • x can be either an exact or an inexact number.
  • For exact numbers, ContinuedFraction[x] can be used if x is rational, or is a quadratic irrational.
  • For quadratic irrationals, ContinuedFraction[x] returns a result of the form {a1,a2,…,{b1,b2,…}}, corresponding to an infinite sequence of terms, starting with the ai, and followed by cyclic repetitions of the bi. »
  • Since the continued fraction representation for a rational number has only a limited number of terms, ContinuedFraction[x,n] may yield a list with less than n elements in this case.
  • For terminating continued fractions, {…,k} is always equivalent to {…,k-1,1}; ContinuedFraction returns the first of these forms.
  • FromContinuedFraction[list] reconstructs a number from the result of ContinuedFraction.

Examples

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Basic Examples  (1)

20 terms in the continued fraction for :

Wolfram Language code: ContinuedFraction[Pi, 20]

Scope  (2)

Rational number:

Wolfram Language code: ContinuedFraction[47 / 17]
Wolfram Language code: 2 + 1 / (1 + 1 / (3 + 1 / 4))
Wolfram Language code: 2 + (1/1 + (1/3 + (1/4)))

Quadratic irrational (recurring continued fraction):

Wolfram Language code: ContinuedFraction[Sqrt[13]]
Wolfram Language code: ContinuedFraction[Sqrt[13], 20]

Generalizations & Extensions  (1)

ContinuedFraction stops when it runs out of precision:

Wolfram Language code: ContinuedFraction[N[Pi]]
Wolfram Language code: ContinuedFraction[N[Pi, 20]]

Applications  (3)

The continued fractions for n^(th) roots of are very regular:

Wolfram Language code: ContinuedFraction[Sqrt[E], 25]

Geometric mean of the first 1000 continued fraction terms in :

Wolfram Language code: N[Apply[Times, ContinuedFraction[Pi, 1000]] ^ (1 / 1000)]
Wolfram Language code: N[Khinchin]

An almost-integer:

Wolfram Language code: ContinuedFraction[Exp[Pi Sqrt[163]], 10]
Wolfram Language code: N[Exp[Pi Sqrt[163]], 50]

Properties & Relations  (2)

FromContinuedFraction is effectively the inverse of ContinuedFraction:

Wolfram Language code: ContinuedFraction[Pi, 20]
Wolfram Language code: FromContinuedFraction[%]
Wolfram Language code: N[%]

Explicit representation using nested fractional parts:

Wolfram Language code: Floor[NestList[1 / FractionalPart[#]&, Pi, 20]]
Wolfram Language code: ContinuedFraction[Pi, 21]

Neat Examples  (1)

Objects showing regularity in their continued fractions:

Wolfram Language code: ContinuedFraction[Coth[1], 20]
Wolfram Language code: ContinuedFraction[BesselI[0, 1] / BesselI[1, 1], 20]
Wolfram Language code: ContinuedFraction[BesselI[1, 2] / BesselI[0, 2], 20]
Wolfram Research (1999), ContinuedFraction, Wolfram Language function, https://reference.wolfram.com/language/ref/ContinuedFraction.html.

Text

Wolfram Research (1999), ContinuedFraction, Wolfram Language function, https://reference.wolfram.com/language/ref/ContinuedFraction.html.

CMS

Wolfram Language. 1999. "ContinuedFraction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ContinuedFraction.html.

APA

Wolfram Language. (1999). ContinuedFraction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ContinuedFraction.html

BibTeX

@misc{reference.wolfram_2026_continuedfraction, author="Wolfram Research", title="{ContinuedFraction}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/ContinuedFraction.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_continuedfraction, organization={Wolfram Research}, title={ContinuedFraction}, year={1999}, url={https://reference.wolfram.com/language/ref/ContinuedFraction.html}, note=[Accessed: 13-June-2026]}