continued fraction
Mathworld





a

0


+









b

1












a

1


+









b

2












a

2


+









b

3












a

3


+
⋱

















{\displaystyle a_{0}+{\cfrac {b_{1}}{a_{1}+{\cfrac {b_{2}}{a_{2}+{\cfrac {b_{3}}{a_{3}+\ddots }}}}}}}


The task is to write a program which generates such a number and prints a real representation of it. The code should be tested by calculating and printing the square root of 2, Napier's Constant, and Pi, using the following coefficients:

For the square root of 2, use 




a

0


=
1


{\displaystyle a_{0}=1}

 then 




a

N


=
2


{\displaystyle a_{N}=2}

. 




b

N




{\displaystyle b_{N}}

 is always 



1


{\displaystyle 1}

.







2


=
1
+








1









2
+








1









2
+








1









2
+
⋱

















{\displaystyle {\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}}


For Napier's Constant, use 




a

0


=
2


{\displaystyle a_{0}=2}

, then 




a

N


=
N


{\displaystyle a_{N}=N}

. 




b

1


=
1


{\displaystyle b_{1}=1}

 then 




b

N


=
N
−
1


{\displaystyle b_{N}=N-1}

.





e
=
2
+








1









1
+








1









2
+








2









3
+








3









4
+
⋱






















{\displaystyle e=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {2}{3+{\cfrac {3}{4+\ddots }}}}}}}}}


For Pi, use 




a

0


=
3


{\displaystyle a_{0}=3}

 then 




a

N


=
6


{\displaystyle a_{N}=6}

. 




b

N


=
(
2
N
−
1

)

2




{\displaystyle b_{N}=(2N-1)^{2}}

.





π
=
3
+








1









6
+








9









6
+








25









6
+
⋱

















{\displaystyle \pi =3+{\cfrac {1}{6+{\cfrac {9}{6+{\cfrac {25}{6+\ddots }}}}}}}




See also

   Continued fraction/Arithmetic for tasks that do arithmetic over continued fractions.


from fractions import Fraction
import itertools
try: zip = itertools.izip
except: pass
 
# The Continued Fraction
def CF(a, b, t):
  terms = list(itertools.islice(zip(a, b), t))
  z = Fraction(1,1)
  for a, b in reversed(terms):
    z = a + b / z
  return z
 
# Approximates a fraction to a string
def pRes(x, d):
  q, x = divmod(x, 1)
  res = str(q)
  res += "."
  for i in range(d):
    x *= 10
    q, x = divmod(x, 1)
    res += str(q)
  return res
 
# Test the Continued Fraction for sqrt2
def sqrt2_a():
  yield 1
  for x in itertools.repeat(2):
    yield x
 
def sqrt2_b():
  for x in itertools.repeat(1):
    yield x
 
cf = CF(sqrt2_a(), sqrt2_b(), 950)
print(pRes(cf, 200))
#1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
 
 
# Test the Continued Fraction for Napier's Constant
def Napier_a():
  yield 2
  for x in itertools.count(1):
    yield x
 
def Napier_b():
  yield 1
  for x in itertools.count(1):
    yield x
 
cf = CF(Napier_a(), Napier_b(), 950)
print(pRes(cf, 200))
#2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516642742746639193200305992181741359662904357290033429526059563073813232862794349076323382988075319525101901
 
# Test the Continued Fraction for Pi
def Pi_a():
  yield 3
  for x in itertools.repeat(6):
    yield x
 
def Pi_b():
  for x in itertools.count(1,2):
    yield x*x
 
cf = CF(Pi_a(), Pi_b(), 950)
print(pRes(cf, 10))
#3.1415926532