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Expressing the Golden Ratio as a Decimal


Date: 03/04/98 at 13:11:02
From: Bryan Willman
Subject: A really difficult extra credit problem

From the equation

   l/w = l + w/l
   
it can be shown that the numerical value of l/w is l + sqrt(5)/2. 
Express the value of l/w, the golden ratio, as a decimal.
      
Thank You Very Much


Date: 03/04/98 at 13:57:46
From: Doctor Rob
Subject: Re: A really difficult extra credit problem

A very effective way to do this is to find the positive root of the
quadratic equation

   w^2 - w - 1 = 0
   
by iteration (Newton's method).

Make a guess w(0), say 2. Then for n = 0, 1, 2, 3, 4, ... compute

     w(n+1) = [w(n)^2 + 1]/[2*w(n) - 1].

This gives you the following table, to 10 decimal places:

     w(0) = 2 = 2.0000000000,
     w(1) = (2^2+1)/(2*2-1) = 5/3 = 1.6666666667,
     w(2) = 34/21 = 1.6190476190,
     w(3) = ...,

and so on. Since w(5) = w(6) to ten decimal places, you only need 5 
steps to get that degree of accuracy. For more accuracy, compute more 
decimal places with each succeeding step. The number of accurate 
decimal places approximately doubles with each succeeding step.

The more straightforward approach is to compute sqrt(5) to many 
decimal places, add 1, and divide by 2. The iteration that works for 
sqrt(5) is

     x(n+1) = [x(n) + 5/x(n)]/2 = [x(n)^2 + 5]/[2*x(n)].

A longhand method of computing square roots is described at the 
following URL:
  http://mathforum.org/dr.math/problems/steve.8.6.96.html   
You can use this to compute sqrt(5) to as many decimal places as you 
want, one digit at a time. The advantage of this method is that it 
involves no rounding, because in this computation you are working with 
integers, and only integers.

-Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Fibonacci Sequence/Golden Ratio

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