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Continued Fractions

Under Construction....


        If we begin with the golden quadratic expression: 0 = x2 + x – 1, we can transform this into the simplest continued fraction imaginable that repeats indefinitely: 
Step 1   0 = x2 + x – 1




Step 2   (0 = x2 + x – 1) + 1




Step 3   (0) + 1 = (x2 + x – 1) + 1




Step 4   1 = x2 + x




Step 5        x
         ------------
         (1 = x2 + x)




Step 6    x       x
         --- = --------
          1    (x2 + x)




Step 7    x         x
         --- = -----------
          1    (x + 1) * x




Step 8    x      1  *  x
         --- = -----------
          1    (x + 1) * x




Step 9    x       1       x
         --- = ------- * ---
          1    (x + 1)    x




Step 10   x       1
         --- = ------- * 1
          1    (x + 1)




Step 11        1
         x = -----
             x + 1




Step 12        1
         x = -----
             1 + x




Step 13              1
         x = -----------------
                 é           ù
                 ê       1   ú
             1 + ê x = ----- ú
                 ê     1 + x ú
                 ë           û




Step 14          1
         x = ---------
                   1
             1 + -----
                 1 + x




Step 15                1
         x = ---------------------
                         1
             1 + -----------------
                     é           ù
                     ê       1   ú
                 1 + ê x = ----- ú
                     ê     1 + x ú
                     ë           û




Step 16            1
         x = -------------
                     1
             1 + ---------
                       1
                 1 + -----
                     1 + x
        In practical (computational) terms, we could have stopped at step 11, input a value of 1 into x on the right-hand side, received a value of 1/2 as the output for x on the left-hand side, re-inserted the result of 1/2 back into x on the right-hand side, and continued until we were satisfied that we had successfully approximated the quadratically, golden proportion of f as: 0.61803398874989.......

        We could have just as easily performed step 2 and beyond a little differently yielding the reciprocal f of: 1.61803398874989....... as its approximation: 

Step 1   0 = x2 + x – 1




Step 2   (0 = x2 + x – 1) – x2




Step 3   (0) – x2 = (x2 + x – 1) – x2




Step 4   – x2 = x – 1




Step 5   x2 = 1 – x

        The continued fraction of: 0 = x2 + x – 1, can be rewritten in short-hand as: x = 1[1,...] to show the repetition of
 
Next Lesson: What does HE Huntley and others have to say on the topic of beauty?

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Monday, 16 August 2004 15:29:58 MST [an error occurred while processing this directive]