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Golden number

With Fibonacci numbers you can calculate the golden ratio as

φ = 1.61803398874989…

 


Explanation

We start with the fibonacci sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...

In a representation you can show squares with sides that correspond to these numbers. This results in rectangles where the ratio between long and short sides develops as 3 : 2, 5 : 3, 8 : 5, ...

1 1 3 8
2
5

If you divide each number in the Fibonacci sequence by its direct predecessor you get

  1 / 1 = 1.000
  2 / 1 = 2.000
  3 / 2 = 1.500
  5 / 3 = 1.666
  8 / 5 = 1.600
  13 / 8 = 1.625
  21 / 13 = 1.615
  34 / 21 = 1.619
  55 / 34 = 1.617
  89 / 55 = 1.618
  144 / 89 = 1.617
  233 / 144 = 1.618
  377 / 233 = 1.618

The divisions stabilize so that the number 1.61803398874989… is created.

 


History

Leonardo of Pisa (1180 - 1241), known as Fibonacci, described the sequence that was later named after him.


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