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

The continued fraction for the Golden Ratio is:

 $\LARGE\phi=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}}}$ 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.

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