The Golden Ratio is a famous and unique ratio that can be found in several different ways. The quickest and easiest way to find it is by dividing a line segment into two smaller segments a and b such that the ratio of a to b is the same as the ratio of segment a+b to segment a. The line segment is said to be cut in extreme and mean ratio (Huntley, 1970). Other names for the Golden Ratio are the Golden Section, the Golden Number, Phi (Φ) and sometimes the Divine Proportion. The ratio dates back to antiquity and has a tendency to pop up in all sorts of unexpected places. It has unique geometrical and algebraic properties, is strongly tied to the Fibonacci numbers, and shows up in nature often. It is also used in design, artwork, architecture and even music because of its propensity to be aesthetically pleasing, and in the case of the latter, pleasing to the ear.
The Golden Ratio is often expressed in the decimal form, 1.61803..., and is an irrational number. The decimal goes on indefinitely, and does not repeat. It was computed to 10 million decimal places in December 1996 via computer (Livio, 2003). The precise value of the Golden Ratio is Φ=(1+√5)/2. This follows easily from the segment division example shared above, taking one of the segments to be 1 and the other to be x, and then using the quadratic formula. The positive root is the Golden Ratio. Interestingly, the negative root Φ=(1-√5)/2 simplifies to - 1/Φ.
