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Continued Fractions


Date: 07/02/97 at 09:47:23
From: Julia MacDonell
Subject: Continued fraction

What exactly IS a continued fraction?

Thanks! 
Jules


Date: 07/02/97 at 09:56:06
From: Doctor Rob
Subject: Re: Continued fraction

Jules,

That is a very interesting question. I'm glad you asked.

A continued fraction is an expression of the form:

             b0
a0 + -----------------------
                b1
     a1 + ------------------
                   b2
          a2 + -------------
                      b3
               a3 + --------
                    a4 + ...

Here all the ai's and bi's are integers, i >= 0.

It may terminate after a finite number of steps. In that case, when
you evaluate it, you will get a rational number (quotient of two
integers). It may be infinite, in the same way that an infinite series
is, or an infinite decimal expansion is. There is a well-developed
theory of continued fractions, especially "simple" ones (bi = 1 for
all i, and ai > 0 for all i > 1). They have many uses. They can be
helpful in approximating irrational numbers, solving Diophantine
equations, finding units in quadratic number fields, and other places.

If you want more information, you can write again, or consult the
following reference:

C. D. Olds, _Continued Fractions_, New Math Library, vol. 9 (1963: New
York, Random House)

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   


Date: 07/03/97 at 20:00:25
From: Julia MacDonell
Subject: expressing continued fractions

How would you express (sqrt A) - B as a continued fraction?

Thank you!


Date: 07/04/97 at 05:49:35
From: Doctor Anthony
Subject: Re: expressing continued fractions

The general expression:

            sqrt(N) + b
            -----------     where N, b, r are integers (N > 0)
                r

can be expressed as a simple recurring continued fraction.

We can write the stages as follows:

  sqrt(N)+b1          sqrt(N)-b2                r2
  ----------  = a1 + -----------    = a1 +  ----------
      r1                 r1                 sqrt(N)+b2

                                                 1
                                    = a1 +  ------------ 
                                            sqrt(N)+b2
                                            -----------
                                                 r2 

a1 is the integral part of [sqrt(N)+b1]/r1 and:

 b2 = a1*r1 - b1              N - b2^2
                        r2 = ----------
                                 r1

Apply these three expressions for calculating a(n), b(n+1), r(n+1) to 
the expression of:

     sqrt(37)+8
     ----------      as a continued fraction.
         9

Here a1 = integral part of expression = 1, also b1 = 8, r1 = 9.

Then b2 = 1 x 9 - 8 = 1     r2 = [37-1]/9 = 4

So the continued fraction at this stage is:

              1
    1  +  ----------
          sqrt(37)+1
          ----------
               4

It is better to lay out the working in tabular form as follows, using 
the expressions for a(n), b(n+1), r(n+1) given above.

     n    1     2      3      4      5       6
    -------------------------------------------
     b    8     1      3      4      5       3
     r    9     4      7      3      4       7
     a    1     1      1      3      2       1

This gives:

 sqrt(37)+8          1     1      1     1      1
 ----------  =  1 + ---- -----  -----  ----  -----  
     9               1+    1+    3+     2+     1+


The example you gave [sqrt(a)-b]/1 is done exactly as shown above, 
with r1 = 1.

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

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