Editing GoldenRatio
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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... 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, 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