## Most recent change of GoldenRatio

Edit made on December 13, 2008 by RiderOfGiraffes at 12:50:37

Deleted text in red / Inserted text in green

WW
Take a piece of rectangular paper, and cut from it the largest possible square.
If the resulting rectangle has the same proportions as the original, then it
was a Golden Rectangle, and its sides were in the Golden Ratio.

The Golden Ratio has the value EQN:(1+\sqrt5)/2, which is about 1.618... Subtracting 1 from the Golden Ratio gives its inverse,
hence EQN:\phi-1=1/\phi. Rearranging we see that EQN:\phi^2-\phi=1 and so EQN:\phi^2-\phi-1=0. Solving this simple quadratic equation
gives two solution, which are EQN:\phi and EQN:1/\phi.

The continued fraction for the Golden Ratio is:

| EQN:\LARGE\phi=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}}}

| From this we can deduce that it is an irrational number. number, since every rational number has a finite continued fraction representation. |

The ratio of successive terms of the Fibonacci sequence approaches the golden ratio,
and the successive truncation of the continued fraction give these ratios.

EQN:\frac{1}{1},\;\frac{2}{1},\;\frac{3}{2},\;\frac{5}{3},\;\frac{8}{5},\;\frac{13}{8},\;\frac{21}{13},\;\frac{34}{21},\;...