The beautiful and useful Fibonacci numbers

Pierre Simon de Laplace (1749-1827) said: “All the effects of Nature are only the mathematical consequences of a small number of immutable laws.”[1] This discovery was confirmed and described by Leonardo Fibonacci. Fibonacci gave an interesting sequence of numbers formed in such a way that the first and second places are occupied by ones, and then each subsequent word equals the sum of the two previous ones.

Leonardo Fibonacci, the mathematical innovator of the thirteenth century, was a solitary flame of the spirit of medieval mathematics. He was born in Pisa, Italy, and for this reason, was known as Leonardo Pizano or Leonardo of Pisa. While his father was the administrator of the district of Bugia on the northern coast of Africa (Bugia is now in Algeria), Leonardo received instruction from a Mauryan master, who introduced him to the Arabic system of counting and methods of calculus.

After extensive travels and study of the calculus system, Fibonacci wrote a work called “Liber Abaci” in 1202, explaining Arabic numbering and how to use it in calculations. This work was an instrument in replacing the clumsy Roman system and introducing a calculus method similar to the one used today. It also included topics on algebra and geometry.

The numbers that make up the Fibonacci sequence are mysterious, although they arise very simply. Their beauty and mystery lie in the fact that they are numbers that can be found in a tree branch or a fir cone, but right next door, a mathematician or a computer scientist directly benefits from these numbers. Indeed, Fibonacci numbers are used in signal coding.


Fibonacci numbers are often found in nature. When a trunk shoots a new branch after a year, it always rests for a year. It’s only after the following year that it gives out a branch. During the first year, we have only the main stem. In the second year, 2 branches, the next year 3, then 5, 8, 13, as in the Fibonacci sequence.

A typical sunflower has a head containing spirals of tightly packed seeds, usually with 34 coils in one direction and 55 coils in the other. Smaller sunflower heads have either 21 and 34 spirals or 13 and 21. One giant sunflower was shown at an exhibition in England. Its spirals were counted. It turned out that they are formed by Fibonacci numbers: 89 and 144.

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This string gives extremely interesting fractional forms when used sequentially in numerators and denominators, namely:


… and so on. Here we are dealing with the technique of chain fractions, which was widespread in the 14th century. At the turn of the twentieth century, this technique is proving indispensable in evaluating the executive efficiency of various and important algorithms.

In addition, the Fibonacci sequence has another interesting property:

If you subtract the product of neighbouring numbers from the square of any number in the Fibonacci sequence, you always get 1, and alternately – once with a plus sign, the other with a minus sign:

The Fibonacci sequence is incredibly interesting. We have cited here only a few of its properties. The string is studied by scientists all the time, especially those associated with the Fibonacci Association (founded in 1963).



  2. Davis Philip J., Hersh Reuten, Świat matematyki, PWN, Warszawa 1994
  3. Iwiński T., Szeregi nieskończone, WSiP, Warszawa 1974
  4. Worobjow N., Liczby Fibonacciego, PWN, Warszawa 1955
  5. Steinhaus H., Kalejdoskop matematyczny, WSiP, Warszawa 1989

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