ON THE GOLDEN RATIO
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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.
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.
참고문헌 (7)
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- Hoggatt, Verner E. Jr., "Fibonacci and Lucas Numbers" (1969), Houghton- Mifflin
- Pattern Recognition: Modular Rings and Integer Structure, "Leyendekkers, J.V.", ~, pp.Raffles KvB Monograph No 9 (2007)
- Leyendekkers, J.V., "The Structure of ‘Pi’. Notes on Number", Theory and Discrete Mathematics, 17(4), pp.61~68 (2011)
- Livio, Mario, "The Golden Ratio" (2002), Golden Books
- Tee, Garry J., "Russian Peasant Multiplication and Egyptian Division in Zeckendorf Arithmetic", Australian Mathematical Society Gazette, 30, pp.267~276 (2003)