Answer to Question #310 in Algebra for kiwi

Question

The golden section is a line segment divided according to the golden ratio: The total length (y+z) is to the longer segment z as z is to the shorter segment y. What is the ratio of the shorter segment y to the longer segment z?

Answer 1

(√5 - 1)/2. Let the segment be divided into segments y and z with (y < z). We have a golden section if (y/z) = (z/(y+z)) = x, where x is a required number. Thus we have an equation for x: x(x + 1) = (y/z) * ((y+z)/z) = 1. Solving it and discarding the negative root we get: x = - (1/2) + (√5/2).