The Most Irrational Number
The most irrational number
The most irrational number turns out to be a number already well known in geometry. It is the number
g = ( + 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 x^{2}  x  1 = 0, so its continued fraction expansion is the simplest of all:
g = $1 + \frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}$
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 
...
Hurwitz' Theorem guarantees the existence of infinitely many convergents with E/M < 1. In this case the oddnumbered convergents must be discarded, and the evennumbered ones are getting as bad as they can be. (In fact this table is evidence that the factor $\sqrt{5}$ 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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