Fibonacci

Contorted fractions.

In the chapter Numerical application Fibonacci we have shown that for the positive zero of the equation x² - x - 1 = 0 there are two very special representations, namely:


x = (1 + (1 + (1 + ...)))

and


They are the simplest continued forms in their kind.
The continous forms suggest that x = (1 + 5)/2 is in a way the 'simplest' irrational number, i.e. the 'simplest' number that cannot be written as a fraction. We will confirm this further on this page.
Usually the number x is called the golden section, and is denoted by the Greek character . The fractions F_n+1/F_n converge to if n approaches infinity. (See: Numerical application Fibonacci)

In the following increasing sequence of (reduced) fractions between 0 and 1 the i-th sequence contains all fractions with a denominator less than or equal to i.
0 1
0 1/2 1
0 1/3 1/2 2/3 1
0 1/4 1/3 1/2 2/3 3/4 1
0 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 1
etc.
These sequences are called Farey sequences. They have unexpected properties. For example: If a/b and c/d are at some time adjacent, then a*d - b*c = -1. The next number to insert between them is (a+c)/(b+d). (An application)
Each time we construct a new Farey sequence we will also construct another sequence called the dyadic sequence. When at some time the fraction (a+c)/(b+d) is inserted between a/b and c/d, then at the same time a new fraction will be inserted in the dyadic sequence, which is the mean of its neighbors.
Thus for the first four rows:
0 1        0 1/2 1         0 1/3 1/2 2/3 1         0 1/4 1/3 1/2 2/3 3/4 1       (Farey)
0 1        0 1/2 1         0 1/4 1/2 3/4 1         0 1/8 1/4 1/2 3/4 7/8 1       (dyadic)
and the fifth row:
0 1/5   1/4 1/3 2/5 1/2 3/5 2/3 3/4   4/5  1       (Farey)
0 1/16 1/8 1/4 3/8 1/2 5/8 3/4 7/8 15/16 1       (dyadic)
The simplest fractions that will not appear in any dyadic sequence are the fractions with denominator 3. The numbers that do not appear in any Farey sequence are the irrational numbers. So, the fractions 1/3 and 2/3 correspond with the simplest irrational numbers.
The numbers F_2n/F_2n+1 from the Farey sequences correspond with the numbers 1/2 + 1/8 + ... + 1/2^2n-1 from the dyadic sequences (For a proof see below). If we take limits then we see that -1 corresponds to 2/3. And also 2- corresponds to 1/3 and (if we do not restrict ourselves to the interval [0,1], to 5/3.
Note that the fraction 1/5 corresponds to 1-(2)/2

Finally we have to proof that F_2m/F_2m+1 corresponds to 1/2 + 1/8 + ... + 1/2^2m-1 (proof).