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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...

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.

| 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, 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},\;...

* http:/www.google.com/search?q=Golden+Ratio