Fibonacci numbers and the Golden Number

If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will find the following series of numbers:

1/1 = 1, 2/1 = 2, 3/2 = 1-5, 5/3 = 1-666..., 8/5 = 1-6, 13/8 = 1-625, 21/13 = 1-61538...

It is easier to see what is happening if we plot the ratios on a graph:

The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. It has a value of approximately 1-61804 , although we shall find an even more accurate value on a_ later page [this link opens a new window] .

• What happens if we take the ratios the other way round i.e. we divide each number by the one following it: 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ..?

Use your calculator and perhaps plot a graph of these ratios and see if anything similar is happening compared with the graph above. You'll have spotted a fundamental property of this ratio when you find the limiting value of the new series!

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

rrrrrrr rrrr^f

The golden ratio 1-618034 is also called the golden section or the golden mean or just the golden number. It is often represented by a greek letter Phi 0. The closely related value which we write as phi with a small "p" is just the decimal part of Phi, namely 0618034.

0 0

Post a comment