1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Continued Fractions

  1. Apr 28, 2012 #1
    Suppose we can write a real number x as a continued fraction like this

    x=a0+1/(a1+1/(a2+1/(a2+...=[a0; a1, a2, a3, a4, ... an...].

    Is there a binary operation f(i,x) so that f(i,x)=ai? I was wondering if there was a formula which gives the ith item in the sequence of integers which is connected to every x in the context of this expansion.

    Every rational number has a unique continued fraction expansion so I think this is a valid question. Moreover, every irrational number has a unique, infinite continued fraction expansion.

    My first guess was to combine the inputs a, b in the Euclidean algorithm from which the continued fraction expansion arises but I don't know how to extract the ith item in the sequence of quotients. Any thoughts?
     
  2. jcsd
  3. Apr 29, 2012 #2
  4. May 1, 2012 #3
    Thanks a lot for answering my question. It's a big step in the right direction.

    My question was how to compute the continued fraction expansion of a fraction. For example, using the Euclidean algorithm, we have
    [tex]
    \frac{7}{10}=0+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3}}}
    [/tex]
    Therefore, $$a_0=0, a_1=1, a_2=3, a_3=3, a_4=0, a_5=0, a_6=0,\dots$$

    Is there a way to compute $$a_n$$ without using the euclidean algorithm?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Continued Fractions
  1. A fraction (Replies: 1)

  2. Continued fractions (Replies: 2)

Loading...