The most irrational number turns out to be a number already well known in geometry. It is the number
which is the length of the diagonal in a regular pentagon of side length 1. This number, known as the "golden mean," has played a large role in mathematical aesthetics. It is not clear whether its supreme irrationality has anything to do with its artistic applications.
The golden mean satisfies the equation x^{2} - x - 1 = 0, so its continued fraction expansion is the simplest of all:
g = 1 + 1 ------ 1 + 1 ------ 1 + 1 ------ 1 + etc.
Its convergents are 1, 2, 3/2, 5/3, 8/5, ... , the ratios of consecutive Fibonacci numbers.
How well are these convergents approximating g? Here are the first few E/M ratios:
convergent E/M c_{1} = 1/1 1.382 c_{2} = 2/1 .8541 c_{3} = 3/2 1.055 c_{4} = 5/3 .9787 c_{5} = 8/5 1.008 c_{6} = 13/8 .9968 c_{7} = 21/13 1.001 c_{8} = 34/21 .9995 ...
So the golden mean can never have a rational approximation as good as 22/7 was for or even as good as 7/5 was for .
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