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The most irrational number turns out to be a number already well known in geometry. It is the number
+ 1)/2 = 1.618033...
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 x2 - 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 c1 = 1/1 1.382 c2 = 2/1 .8541 c3 = 3/2 1.055 c4 = 5/3 .9787 c5 = 8/5 1.008 c6 = 13/8 .9968 c7 = 21/13 1.001 c8 = 34/21 .9995 ...
Hurwitz' Theorem guarantees the existence of infinitely many
convergents with E/M < 1. In this case the odd-numbered
convergents must be discarded, and the even-numbered ones are
getting as bad as they can be. (In fact this table is evidence
that the factor in Hurwitz'
theorem cannot be improved!)
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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