
Re: Can a fraction have none noneending and nonerepeating decimal representation?
Posted:
Aug 2, 2013 2:44 AM


On 08/01/2013 08:25 PM, Richard Tobin wrote: > In article <d7dd1b23452d4ecfa5c0fa880d9dad76@googlegroups.com>, > <jonas.thornvall@gmail.com> wrote: > >> And i may claim that Pi is a rational, because i know i can write it as >> a continued fraction. > > But that isn't what rational means. > >  Richard >
This made me think of what happens if we try to apply the Euclidean Algorithm to sticks of length (1+sqrt(5))/2 and 1.
Of course, (1+sqrt(5))/2 = phi = 1.618.... . Then, phi 1 = 0.618.... = 1/phi.
So, after chopping off 1 unit from the stick of length phi, it now has length 1/phi, and the two sticks have lengths in ratios 1 :: 1/phi or phi :: 1.
phi = 1+ 1/(1+1/1+1/(1+ (to suggest the continued fraction expansion).
Yet, phi is irrational.
david
 On Hypnos, http://messagenetcommresearch.com/myths/bios/hypnos.html

