The Golden Section as was stated earlier is the division of a line segment into its extreme and mean ratio. This most basic way of constructing the Golden Ratio gives us enough to precisely solve for it. If we take the smaller section to be 1 and the larger to be x we have the following proportion 1/x = x /(x+1) If we solve this we get (x+1) = x2 or x2 - x - 1 = 0 and then employing the quadratic formula we get x = (1±√5) / 2 . Since we are talking about distance, we take the positive root and this is the Golden Ratio. This construction was shown in Euclid's books (Huntley, 1970).
Two other surprising mathematical manifestations of the Golden Ratio are in nested square-roots and fractions involving 1. The first is 
which looks impossible to solve as it goes on to infinity. If we set it equal to x we have x= . If we square both sides we have x2 = 1+ and since the expression on the right hand side after 1+ is equal to x, we again have x2 = 1 + x which gives us the Golden Ratio as shown above (Livio, 2003).
The second is the unending fraction
Again if we let x = then we see that the denominator of the right side is equal to x and thus we have x = 1 + 1/x which again simplifies to x2 = x+1 (Livio, 2003).