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Continued Fractions


Date: 10/07/97 at 16:15:35
From: Sarah
Subject: Continued fractions

Dear Dr. Math,

I was recently assigned a project in math class in which I must 
present to the class a lesson on continued fractions. Up until this 
point I have not been able to figure out exactly what "continued 
fractions" are. I have searched the Net and also a few math textbooks 
and I have not found even the simplest definition of the fractions.  
I would appreciate you sharing ANY knowlegde that you have on 
continued fractions.  

Thank you,

Sarah


Date: 10/10/97 at 11:53:05
From: Doctor Luis
Subject: Re: Continued fractions

A continued fraction is an expression of the form,

  A = a_0 + 1/c_0

where

  c_0 = a_1 + 1/c_1

  c_1 = a_2 + 1/c_2

   .
   .                   (0 denominators are not allowed, of course)
   .

If the process stopped at, say,

  c_n = a_(n+1) + 1/c_(n+1)    
 
then we would have a finite continued fraction.

Now, the initial question is whether the process converges in the 
infinite case.

If a_0 is an integer and the rest of the a's are positive integers,
the process converges to a real number; and all real numbers can be 
represented in this way. Simple (or ``regular'', as some authors 
write) continued fractions are those with these requirements on 
a_0, a_1, a_2, ....


It is standard to use [a_0,a_1,a_2,a_3,...] to represent

a =  a_0 +      1
           --------------
           a_1 +    1
                 ---------
                 a_2 +  1
                      ----
                      a_3 + ...


There's an interesting way you can find a continued fraction
representation of 1 + the square root of 2.

              x   = 1 + sqrt(2)

            x - 1 = sqrt(2)

     x^2 - 2x + 1 = 2

             x^2  = 2x + 1 

                         1
      or     x    = 2 + ---
                         x 
  
    "Substituting" for x in the lefthand side

                           1
             x   =  2 + --------
                             1
                        2 + ---  
                             x 

    Repeating this process an infinite number of times you'll
    get the infinite continued fraction,

                              1
             x  =  2 + -------------------
                                1
                       2 + ----------------
                                   1 
                           2 + -----------
                                     1                                  
                               2 + -------
                                   2 + ...


   Using the notation I introduced above, we have

     1+sqrt(2) = [2,2,2,2,2,....]

  Note that this gives us the continued fraction representation
  of the square root of 2, namely

       sqrt(2) = [1,2,2,2,2,....]

  Which is interesting because, if you remember, the sqrt(2)
  has a nonterminating and nonrepeating (since it's irrational)
  decimal representation 1.414213562373...... BUT it has the
  repeating continued fraction representation [1,2,2,2,...]

  (Remember that rational numbers have repeating decimal
   representations, like:

   1/3 = 0.3333333.....

   2/7 = 0.285714285714285714.......
    
   1/1 = 0.99999999999......  

   and so on )

Why don't you try to solve the following?

 Problem: Show that the continued fraction for sqrt(n(n+1)) is          
          [n,2,2n,2,2n,2,2n...].

You should be able to find the theory of continued fractions in 
(almost) any elementary or standard number theory textbook. I 
certainly encourage you to find out more about this fascinating 
subject.

Here are a couple of links I found that you might find interesting:

Continued Fraction Representations involving e

   http://pauillac.inria.fr/algo/bsolve/constant/e/cntfrc.html   

Continued Fractions Involving Pi

   http://www.mathsoft.com/asolve/constant/pi/frc.html   

 I hope this helped :)

-Doctor Luis,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory
High School Sequences, Series

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