On the golden ratio

Abstract

The structure of the decimal expansion of the Golden Ratio is examined in decimal and modular forms through the use of various properties of the Fibonacci numbers, particularly the roots of the associated polynomial and the golden ratio. While the ratio Fn+1/Fn approaches the golden ratio it cannot have both terms even, whereas the ratio Fn+6/Fn can. The decimal string of the golden ratio is given in ratio and binomial forms and analysed with the modular ring Z4 and the sequential structure. The decimal part of the golden ratio is also related to pi.