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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 = (sqr5 + 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 = $1 + \frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}$

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
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 $\sqrt{5}$ in Hurwitz' theorem cannot be improved!)

So the golden mean can never have a rational approximation as good as 22/7 was for pi or even as good as 7/5 was for sqr2.

 

 


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