## Pi and Phi

Euclid and the Golden Ratio looked at Euclid’s algorithm and conitnued fractions. Rational numbers have finite number of steps to Euclid’s algorithm and hence a finite continued fraction representation.

The more abundant irrational numbers have infinite continued fraction representations and the arbitrary length of representation a continued fraction of a irrational leads to a rational approximation of the particular irrational that is *best* from the view point of rational number with the same denominator.

Two interesting irrational numbers are and .

The following animated gif explores the continued fraction representation of :

illustrates the simplest of repeating infinite continued fraction representation. Using the list notation and {a} to denote the repeated sequence:

The following illustrates successive approximations:

As can be seen, the continued fraction approximations are generated from the where is the nth Fibonacci number.

As has been observed: