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 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 or even as good as 7/5 was for
.
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