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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.
  • 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.
EXAMPLESCLOSE ALL
Basic Examples   (1)
20 terms in the continued fraction for Pi:
20 terms in the continued fraction for Pi:
In[1]:=
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Out[1]=
Scope   (2)
Rational number:
Quadratic irrational (recurring continued fraction):
Generalizations & Extensions   (1)
ContinuedFraction stops when it runs out of precision:
Applications   (3)
The continued fractions for n^(th) roots of e are very regular:
Geometric mean of the first 1000 continued fraction terms in pi:
An almost-integer:
Properties & Relations   (2)
FromContinuedFraction is effectively the inverse of ContinuedFraction:
Explicit representation using nested fractional parts:
Neat Examples   (1)
Objects showing regularity in their continued fractions:
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