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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}

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Categories: Mathematica, Mathematics
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