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/
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