Fibonacci Numbers and the Golden Ratio: Unlocking the Secrets of √5

                                                                                                                           By Vishnu Vinjamuri

Abstract:

The relationship between the Fibonacci sequence and the golden ratio, particularly in relation to the number 5, illustrates how the irrational number can be approximated through simple arithmetic involving Fibonacci numbers. As the sequence progresses, an accuracy of up to 15 digits is attainable, a level of precision commonly utilized in applications such as Microsoft Excel. By leveraging the error ratio generated at each stage, a weighted average of the results from subsequent stages is computed, facilitating faster convergence. The outcomes are not only promising but also intriguing, with convergence achieved at a Fibonacci number as low as 6765. 

Introduction:

Fibonacci Sequence and Fibonacci Numbers:

It is well known that Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted by F_n.

The list of Fibonacci numbers include:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144,233,377,610,987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, .......

Golden Ratio:

The Golden Ratio, commonly denoted by the Greek letter phi (φ), is a significant mathematical constant with an approximate value of 1.6180339887. It is algebraically defined as the positive solution to the equation:                                   

- Equation 1

The positive solution to Equation 1 is, .

Fibonacci Sequence and Golden Ratio:

The ratio of two consecutive Fibonacci numbers approaches the golden ratio as the numbers increase towards infinity.

This is represented by the following equation:

                                      -Equation 2

Equation 2 can be re-written as:

Which means, for a large n,

                                        - Equation 3

To obtain the value of  , Equation 3 can be re-written as:

-1                                     -Equation 4

Using Equation 4, we can develop approximations for

Computation Using the Known Relation:

Let us start with the number 3, and present the results in the below table.

We will compare the results with the established value of 2.23606797749979. In the Python program, an accuracy of up to 32 digits is implemented for reasons that will be elaborated upon later.

The results are presented in the below Table 1.

Table 1 – Table 1 – Estimation of  from the known relation between  and Fibonacci numbers

Fibonacci Number	Estimate	% error	Error Ratio
0
1
1
2
3	3.000000000000000	34.1640786499874%
5	2.000000000000000	-10.5572809000084%	-30.9017%
8	2.200000000000000	4.3498389499900%	-37.0820%
13	2.250000000000000	-1.6130089900093%	-38.6271%
21	2.230769230769230	0.6230589874905%	-38.0329%
34	2.238095238095230	-0.2369671577017%	-38.2592%
55	2.235294117647050	0.0906618499902%	-38.1727%
89	2.236363636363630	-0.0346080647157%	-38.2057%
144	2.235955056179770	0.0132222663539%	-38.1931%
233	2.236111111111110	-0.0050499949535%	-38.1979%
377	2.236051502145920	0.0019289937405%	-38.1961%
610	2.236074270557020	-0.0007368002241%	-38.1968%
987	2.236065573770490	0.0002814340751%	-38.1965%
1597	2.236068895643360	-0.0001074980423%	-38.1966%
2584	2.236067626800250	0.0000410606287%	-38.1966%
4181	2.236068111455100	-0.0000156837602%	-38.1966%
6765	2.236067926333410	0.0000059906636%	-38.1966%
10946	2.236067997043600	-0.0000022882301%	-38.1966%
17711	2.236067970034710	0.0000008740258%	-38.1966%
28657	2.236067980351190	-0.0000003338485%	-38.1966%
46368	2.236067976410650	0.0000001275185%	-38.1966%
75025	2.236067977915800	-0.0000000487078%	-38.1966%
121393	2.236067977340880	0.0000000186045%	-38.1966%
196418	2.236067977560480	-0.0000000071067%	-38.1966%
317811	2.236067977476600	0.0000000027141%	-38.1966%
514229	2.236067977508640	-0.0000000010371%	-38.1966%
832040	2.236067977496400	0.0000000003958%	-38.1966%
1346269	2.236067977501080	-0.0000000001516%	-38.1966%
2178309	2.236067977499290	0.0000000000577%	-38.1966%
3524578	2.236067977499970	-0.0000000000223%	-38.1966%
5702887	2.236067977499710	0.0000000000081%	-38.1966%
9227465	2.236067977499810	-0.0000000000036%	-38.1966%
14930352	2.236067977499770	0.0000000000009%	-38.1966%
24157817	2.236067977499790	-0.0000000000009%	-38.1966%

 

The error ratio is negative due to the oscillation of the estimate around its true value, transitioning from one Fibonacci number to the next.

An accuracy of fifteen digits is attained when F_n+1=24157817 and F_n=14930352. In this context, Equation 4 can be reformulated as:

                              - Equation 5                           

The Fibonacci sequence is also formed by Binet’s formula which is as follows:

                               - Equation 6

It can be inferred and demonstrated that the error estimation  using Equation 4 decreases as  for large n, which is approximately equal to 38.1966% as determined in Table 1.

The Excel-based error estimates were insufficient for accurately producing the aforementioned error for large Fibonacci numbers due to limitations in the number of decimal places in Excel.

Consequently, the calculations were refined using Python programming to achieve an accuracy level of 32 decimal places before being imported back into the Excel table.

Optimizing Error Reduction and Enhanced Solutions:

Given that the errors decrease by approximately 38.1966% from one step to the next, we shall pursue enhanced convergence by employing a weighted average of the solution between successive steps.

We will designate the initial solution from Equation 4 at the n^th step as r5_n,i.

By calculating the weighted average based on the error ratio, the adjusted value of the solution will be presented as below:

      - Equation 7

Where r5_n,w is the modified estimate of  as weighted average of the solution from Equation 4.

The findings from Equation 7 are detailed in Table 2 below. It is evident that the convergence is significantly more rapid, occurring at a value as low as 6765, in contrast to 24157817 as indicated in Table 1.

 Table 2 – Optimized Error Reduction Solution

Fibonacci Number	Modified Estimate for	% error
0
1
1
2
3	2.27639319635939	1.8033986115524%
5	2.24120226788020	0.2296124461366%
8	2.23685242618125	0.0350816115322%
13	2.23618034018203	0.0050250119125%
21	2.23608448454537	0.0007382175206%
34	2.23607037951385	0.0001074213345%
55	2.23606832828112	0.0000156874179%
89	2.23606802866699	0.0000022882670%
144	2.23606798496276	0.0000003337543%
233	2.23606797858983	0.0000000487482%
377	2.23606797765834	0.0000000070907%
610	2.23606797752311	0.0000000010428%
987	2.23606797750312	0.0000000001490%
1597	2.23606797750030	0.0000000000229%
2584	2.23606797749985	0.0000000000029%
4181	2.23606797749980	0.0000000000006%
6765	2.23606797749979	0.0000000000000%

Analysis of Table 2 indicates that the error decreases by approximately 685%. Further calculations involving weighted averages may introduce unnecessary complexity.

Conclusions and Inferences:

• The relationship between and Fibonacci numbers is well-established, allowing us to express with an accuracy of up to 15 decimal places as  .                            
• By utilizing the weighted average of estimates derived from the relationship between and Fibonacci numbers, we can similarly express  with an accuracy of up to 15 decimal places as .
• Enhanced accuracy can be achieved by selecting larger Fibonacci numbers.