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Continued Fraction

Date: 09/18/97 at 04:30:58
From: brian MacDonell
Subject: Continued fraction

How would you express (square root of 3) - 1 as a continued fraction?

Date: 09/24/97 at 12:24:21
From: Doctor Rob
Subject: Re: Continued fraction

You need to know the general way to expand a quadratic surd as a 
continued fraction. 

In general, let t(0) be any number you want to expand.  Then for
each n >= 0, let

     a(n) = [t(n)]

(greatest integer less than or equal to t(n)), and let

   t(n+1) = 1/(t(n)-a(n))

Continue this until either the denominator is zero (and the process 
ends), or until t(k) = t(i) for some k and i (and the process is 
obviously periodic). Then the continued fraction of t(0) is given by

t(0) = a(0) + ------------------------ = [a(0); a(1), a(2), a(3), ...]
              a(1) + -----------------
                     a(2) + ----------
                            a(3) + ---

When t(0) is a quadratic surd, it can be put into the form
(P + R*Sqrt[D])/Q, where P, Q, R, and D are integers, and D > 0 
and not divisible by a square, and Q is a divisor of P^2 - R^2*D.  
When you subtract the integer part a(0), you get a quantity of the 
same form (but different P), and satisfying the same conditions.  
When you take the reciprocal, and then rationalize the denominator, 
you get another quantity of the same form (but different P and Q, 
and possibly R), and satisfying the same conditions.

In the case at hand, P = -1, Q = 1, R = 1, and D = 3, when you begin.  
The values of t(n) and a(n) can be tabulated:

   n    t(n)         a(n)    t(n)-a(n)
   0  (Sqrt[3]-1)/1   0    (Sqrt[3]-1)/1
   1  (Sqrt[3]+1)/2   1    (Sqrt[3]-1)/2
   2   ...           ...    ...

You continue until t(n) = t(i) for some i < n.  Then you will be able 
to read off the continued fraction expansion of Sqrt[3] - 1.

-Doctor Rob,  The Math Forum
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Associated Topics:
High School Sequences, Series

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