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ContinuedFraction

ContinuedFraction
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 corresponds to the expression .
  • 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 , corresponding to an infinite sequence of terms, starting with the , and followed by cyclic repetitions of the . »
  • Since the continued fraction representation for a rational number has only a limited number of terms, ContinuedFraction may yield a list with less than n elements in this case.
  • For terminating continued fractions, is always equivalent to ; ContinuedFraction returns the first of these forms.
20 terms in the continued fraction for :
20 terms in the continued fraction for :
In[1]:=
Click for copyable input
Out[1]=
Rational number:
Quadratic irrational (recurring continued fraction):
ContinuedFraction stops when it runs out of precision:
The continued fractions for n^(th) roots of are very regular:
Geometric mean of the first 1000 continued fraction terms in :
An almost-integer:
FromContinuedFraction is effectively the inverse of ContinuedFraction:
Explicit representation using nested fractional parts:
Objects showing regularity in their continued fractions:
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