The Remarkable Relationship Between the Fibonacci Numbers and the Golden Ratio
You can derive the Golden Ratio from the Fibonacci numbers and you can generate the Fibonacci numbers from the Golden Ratio
The Golden Ratio — also know as the Divine Constant — is a fascinating natural constant. It occurs across many fields of mathematics and science and can be represented in many ways. The easiest way to describe the Golden Ratio is to take a line segment and divide it into two sections, such that the ratio of the entire line segment to the larger section is the same as the ratio of the larger section to the smaller section. That ratio is the Golden Ratio. This picture may help:
One of the most common way of representing The Golden Ratio is the unique positive solution to this equation:
Some simple algebraic manipulation and the use of the formula for quadratic roots renders the golden ratio to be (1 + √5)/2. This is an irrational number approximately equal to 1.618.
There are many other ways the Golden Ratio can be represented. One commonly known approach is the limit of the ratios of consecutive Fibonacci numbers. Recall the the Fibonacci sequence is a sequence where the first two terms are 1 and then each subsequent term is the sum of the two previous terms. So
The first few terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and as the sequence progresses, the ratio of any term and its preceding term approaches the golden ratio. That is:
Recently I tried quite a challenging high school examination question which led me to see that, not only does the golden ratio arise from the Fibonacci numbers, but by deriving the golden ratio using other methods, the Fibonacci numbers naturally make an appearance. I found this quite fascinating.