Irrational Number 4

$e-MATH$
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 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.

There is a lower bound for the error in approximation of an irrational number by its convergents:

If the denominator of the n-th convergent is q_n, then the error in approximation by the n-th convergent is at least 1/q_n*(q_n+q_n+1).

This means that the quantity E/M we tabulated is always at least sqr5*q_n/(q_n+q_n+1). In the case of our number g, since q_n is the n-th Fibonacci number, this fraction has denominator q_n+2, and we may rewrite the fraction as sqr5*q_n/q_n+2 = sqr5*(q_n/q_n+1)(q_n+1/q_n+2) = sqr5/(c_n*c_n+1). As n gets large the convergents approach g and the lower bound on E/M approaches sqr5/g² = .854. So g can never have a rational approximation as good as 22/7 was for pi.

The actual E/M's are larger: 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. This justifies g being called the most irrational number.