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## 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 $\pi$ and $\phi$.

The following animated gif explores the continued fraction representation of $\pi$:

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

$\phi =\frac{1+\sqrt{5}}{2}=[1,\{1\}]$

The following illustrates successive approximations:

As can be seen, the continued fraction approximations  are generated from the $\frac{F_{n+1}}{F_n}$ where $F_n$ is the nth Fibonacci number.

As has been observed:

$1+\sqrt{2}=[2,\{2\}]\neq 2 [1,\{1\}]=1+\sqrt{5}$