Fibonacci and Golden Ratio

But that gets back to my original answer, "The Golden Ratio is an irrational number, and thus an infinite number. It is not possible to compute its decimal expansion in a finite amount of time."

Jan 2011-8-17

Fortunately the universe is finite. Therefore I do not believe, that an infinite number will match into it. While there is a minimal Planck length and a minimal Plank time, I propose a Planck eps for irrational numbers. According to Rupert Sheldrake, I claim that PI has as many numbers as has been calculated already. And after reading http://scientopia.org/blogs/goodmath/2010/12/08/really-is-wrong/ I'm not sure at all anymore about this fuzzy digits stuff.

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Answer 2

Walter Roberson 2011-4-4

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The Golden Ratio is an irrational number, and thus an infinite number. It is not possible to compute its decimal expansion in a finite amount of time.

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Sean de Wolski 2011-4-4

Soya sausages? That's like one term in the Taylor-series expansion of sausages.

Walter Roberson 2011-8-17

Jan, Soya Beans used for the production of soya products are the dried fruit of the soya plant, and thus were not covered by the Veggi-Toolbox in R2011a (which, I understand, is still withheld from production due to legal battles over whether Tomatoes are fruits or vegetables....)

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Answer 3

Walter Roberson 2011-4-4

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Let F(t) be Fibonacci number #t. Then

y = 100;  %initial guess
x = (F(t+2) * y + F(t+1)) / (F(t+1) * y + F(t));
while x ~= y;
  y = x;
  x = (F(t+2) * y + F(t+1)) / (F(t+1) * y + F(t));
end

When the loop finishes (no more than a few centuries later, I'm sure), x and y will be the Golden ratio.

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Walter Roberson 2011-4-4

Not completely certain. It worked for the F() values that I tried.

Jack Lê 2011-8-17

Thanks

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Answer 4

Kishore 2023-7-8

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fib=[0 1];
i=3;
while(i<=21)
    fib(i)=fib(i-1)+fib(i-2);
    gr=fib(i)/fib(i-1)
    i=i+1;
end
gr = 1
gr = 2
gr = 1.5000
gr = 1.6667
gr = 1.6000
gr = 1.6250
gr = 1.6154
gr = 1.6190
gr = 1.6176
gr = 1.6182
gr = 1.6180
gr = 1.6181
gr = 1.6180
gr = 1.6180
gr = 1.6180
gr = 1.6180
gr = 1.6180
gr = 1.6180
gr = 1.6180
disp(fib)
  Columns 1 through 16

           0           1           1           2           3           5           8          13          21          34          55          89         144         233         377         610

  Columns 17 through 21

         987        1597        2584        4181        6765

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Answer 5

Guna 2024-4-16，9:47

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% Function to calculate Fibonacci sequence up to a certain number of terms
function fib_sequence = fibonacci(n)
    fib_sequence = zeros(1, n);
    fib_sequence(1) = 0;
    fib_sequence(2) = 1;
    for i = 3:n
        fib_sequence(i) = fib_sequence(i-1) + fib_sequence(i-2);
    end
end
% Calculate the golden ratio using Fibonacci sequence
n = 20; % Number of Fibonacci terms to generate
fib_seq = fibonacci(n);
% Calculate the ratio of consecutive Fibonacci numbers
golden_ratio_approximations = fib_seq(3:end) ./ fib_seq(2:end-1);
% Display the approximations of the golden ratio
disp('Approximations of the golden ratio using Fibonacci sequence:');
Approximations of the golden ratio using Fibonacci sequence:
disp(golden_ratio_approximations);
    1.0000    2.0000    1.5000    1.6667    1.6000    1.6250    1.6154    1.6190    1.6176    1.6182    1.6180    1.6181    1.6180    1.6180    1.6180    1.6180    1.6180    1.6180