document.write( "Question 65951: I am looking for someone who would know about the \"golden mean\"?? I am guessing that, when evaluating complex fractions, there is a pattern that would lead you to the \"golden mean\"?? If anyone can help me answer this question, I would appreciate it. \n" ); document.write( "
Algebra.Com's Answer #46716 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
I am looking for someone who would know about the \"golden mean\"?? I am guessing\r\n" );
document.write( "that, when evaluating complex fractions, there is a pattern that would lead you\r\n" );
document.write( "to the \"golden mean\"?? If anyone can help me answer this question, I would\r\n" );
document.write( "appreciate it.\r\n" );
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document.write( "Articles on \"the golden mean\", \"the golden section\", and \"the golden ratio\" are\r\n" );
document.write( "all referring to the same thing.  You can find lots of articles on the\r\n" );
document.write( "internet, using Google, about this golden \"thing\".  I prefer to call it the\r\n" );
document.write( "golden ratio, because it really is a ratio. The golden ratio can be found in\r\n" );
document.write( "art and nature. \r\n" );
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document.write( "If two numbers x and y are such that x is to y as y is to x+y, then\r\n" );
document.write( "they are said to be in the golden ratio.\r\n" );
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document.write( "Suppose x and y are in golden ratio.\r\n" );
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document.write( " x       y\r\n" );
document.write( "——— = ———————\r\n" );
document.write( " y     x + y\r\n" );
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document.write( "x(x + y) = yē\r\n" );
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document.write( "xē + xy = yē\r\n" );
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document.write( "xē + yx - yē = 0\r\n" );
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document.write( "Use the quadratic formula:\r\n" );
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document.write( "Use the quadratic formula:\r\n" );
document.write( "                  ______ \r\n" );
document.write( "            -b ą Öbē-4ac\r\n" );
document.write( "        x = —————————————\r\n" );
document.write( "                2a \r\n" );
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document.write( "where a = 1; b = y; c = -yē\r\n" );
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document.write( "                     _____________ \r\n" );
document.write( "             -(y) ą Ö(y)ē-4(1)(-yē)\r\n" );
document.write( "        x = ————————————————————————\r\n" );
document.write( "                      2(1) \r\n" );
document.write( "                   _____ \r\n" );
document.write( "             -y ą Öyē+4yē\r\n" );
document.write( "        x = ————————————\r\n" );
document.write( "                 2\r\n" );
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document.write( "                   ___ \r\n" );
document.write( "             -y ą Ö5yē\r\n" );
document.write( "        x = ——————————\r\n" );
document.write( "                 2\r\n" );
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document.write( "                    __\r\n" );
document.write( "             -y ą yÖ5\r\n" );
document.write( "        x = —————————\r\n" );
document.write( "                2\r\n" );
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document.write( "Factor out y\r\n" );
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document.write( "                     _\r\n" );
document.write( "             y(-1 ą Ö5)\r\n" );
document.write( "        x = —————————\r\n" );
document.write( "                 2\r\n" );
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document.write( "We will only use the + symbol, because the - sign\r\n" );
document.write( "gives a negative number.\r\n" );
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document.write( "                     _\r\n" );
document.write( "             y(-1 + Ö5)\r\n" );
document.write( "        x = —————————\r\n" );
document.write( "                 2\r\n" );
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document.write( "Divide both sides by y\r\n" );
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document.write( "                     _\r\n" );
document.write( "       x      -1 + Ö5\r\n" );
document.write( "      ——— = —————————\r\n" );
document.write( "       y         2\r\n" );
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document.write( "So the right side is the golden ratio.\r\n" );
document.write( "                  \r\n" );
document.write( "The golden ratio is .6180339887\r\n" );
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document.write( "Notice that this number is the only number whose\r\n" );
document.write( "reciprocal can be found by simply adding 1.\r\n" );
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document.write( "The reciprocal of .6180339887 is 1.6180339887\r\n" );
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document.write( "Each of the 5 concruent isosceles triangles forming\r\n" );
document.write( "the 5 points of an ordinary 5-pointed star like this:\r\n" );
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document.write( "«\r\n" );
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document.write( "are such that each of their two equal sides are\r\n" );
document.write( "in golden ratio with its base.\r\n" );
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document.write( "If you take the Fibbonacci sequence which begins\r\n" );
document.write( "with the first two terms being 1 and each term\r\n" );
document.write( "beginning with the third consisting of the sum\r\n" );
document.write( "of the two preceding terms, i.e.,\r\n" );
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document.write( "1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144\r\n" );
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document.write( "have these successive ratios:\r\n" );
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document.write( "1/1 = 1, which is more than the golden ratio, .6180339887\r\n" );
document.write( "1/2 = .5, which is less than the golden ratio, .6180339887\r\n" );
document.write( "2/3 = .6666666667, which is more than the golden ratio, .6180339887\r\n" );
document.write( "3/5 = .6, which is less than the golden ratio, .6180339887\r\n" );
document.write( "5/8 = .625, which is more than the golden ratio, .6180339887\r\n" );
document.write( "8/13 = .6153846154, which is less than the golden ratio, .6180339887\r\n" );
document.write( "13/21 = .619047619, which is more than the golden ratio, .6180339887\r\n" );
document.write( "21/34 = .6176470588, which is less than the golden ratio, .6180339887\r\n" );
document.write( "34/55 = .6181818182, which is more than the golden ratio, .6180339887\r\n" );
document.write( "55/89 = .6179775281, which is less than the golden ratio, .6180339887\r\n" );
document.write( "89/144 = .6180555556, which is more than the golden ratio, .6180339887\r\n" );
document.write( "etc., etc., etc.\r\n" );
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document.write( "Notice how the ratios go above then below the golden ratio,\r\n" );
document.write( "but getting closer and closer to it each time.  The ratios will\r\n" );
document.write( "never reach the golden ratio exactly, but will eventually become\r\n" );
document.write( "closer than any tolerance you choose.  \r\n" );
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document.write( "Edwin
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