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The golden ratio is the ratio such that the whole portion to the larger portion equals the larger portion to the lesser, or in other words, we have that a/b=b/c when a=b+c.
(a/b) = (b/c) when a = b + c (a/b) = (b/c) ac = b^2 a = b + c c = a - b a(a - b) = b^2 a^2 - ab = b^2 a^2 - ab - b^2 = 0 (a^2/b^2) - (a/b) - 1 = 0 the last equation is a quadratic in a/b. (a^2/b^2) - (a/b) = 1 (a^2/b^2) - (a/b) + (1/4) = (5/4) ((a/b) - (1/2))^2 = (5/4) (a/b) - (1/2) = (sqrt5)/2 a/b = (sqrt5 + 1)/2
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