Please correct your erroneous statement that "Some elementary algebra shows that in this case the ratio of AC to CB is equal to the irrational number 1.618 (precisely half the sum of 1 and the square root of 5)". 1.618 is not an irrational number, nor is it equal to the golden ratio. A better statement would be ""Some elementary algebra shows that in this case the ratio of AC to CB is equal to an irrational number approximately equal to 1.618 (the ratio is precisely half the sum of 1 and the square root of 5)"