## Theory of Fibonacci numbers

The Fibonacci mathematician lived in the twelfth century and was one of the most famous scientists of his time. Among his achievements is the introduction of Arabic numerals in Europe in place of Roman numerals. He opened the summation sequence, called his name:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

This mathematical sequence arises when starting from 1, 1 the next number is obtained by adding the two previous ones. But why is this sequence so important?

This sequence asymptotically (approaching more and more slowly and slowly) tends to some constant relation. However, this ratio is irrational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It is impossible to express it accurately. If any member of the Fibonacci sequence is divided into the preceding one (for example, 13: 8), the result is a value oscillating around the irrational value 1.61803398875 … and once more superior, it does not reach it. But even after spending an eternity on this, it is impossible to know the ratio exactly, up to the last decimal figure. For the sake of brevity, we will quote it as 1.618.

**The Fibonacci sequence has very interesting features, not the last of which is an almost constant relationship between numbers:**

1. The sum of any two adjacent numbers is equal to the next number in the sequence. For example: 3 + 5 = 8, 5 + 8 = 13 and so on.

2. The ratio of any number of sequence to the next approaches 0.618 (after the first four numbers).

For example: 1/1 = 1.00; 1/2 = 0.50; 2/3 = 0.67; 3/5 = 0.60; 5/8 = 0.625; 8/13 = 0.615; 13/21 = 0.619 and so on.

Notice how the values of the ratios oscillate around the value 0.618, and the range of fluctuations gradually narrows; as well as the values: 1.00; 0.50; 0.67.

3. The ratio of any number to the previous one is approximately equal to 1.618. This is the reciprocal of 0.618.

For example: 13/8 = 1.625; 21/13 = 1.615; 34/21 = 1.619. The higher the numbers, the more they approach the values of 0.618 and 1.618.

4. The ratio of any number to the next one after it, s close to 0.382, and to the previous one through one – to 2.618.

For example: 13/34 = 0.382, 34/13 = 2.615. Another important fact is that the square of any Fibonacci number is equal to the number in the sequence before it multiplied by the number after it, plus or minus 1.

5 ^{2} = (3 x 8) + 1

8^{ 2} = (5 x 13) – 1

13 ^{2} = (8 x 21) + 1

Plus and minus constantly alternate. This is another manifestation of an integral part of the Elliott wave theory, called the rule of alternation. It says that complex correction waves alternate with simple ones, strong impulse ones are full – with weak corrective waves and so on.

The Fibonacci sequence contains other curious relationships, or coefficients, but the ones that we just brought out are the most important and famous. In fact, Fibonacci is not the pioneer of these numbers. The fact is that the coefficients 1.618 or 0.618 were known to ancient Greek and ancient Egyptian mathematicians, who called them the “golden ratio” or “golden section”. His traces we find in music, fine arts, architecture and biology. The Greeks used the principle of “golden section” in the construction of the Parthenon, the Egyptians – the Great Pyramid in Giza. Properties of the “golden ratio” were well known to Pythagoras, Plato and Leonardo da Vinci; These coefficients are widely used today.

Individual numbers from the Fibonacci summation sequence can be seen in the movement of commodity prices. Oscillations of ratios near 1.618 to a greater or lesser magnitude are found in the Elliott wave theory, where they are described by the “rule of alternation”. A person subconsciously searches for the Divine proportion: it is needed to satisfy his need for comfort.

It was the number 0.618 that became the basis for application in the technical analysis of Fibonacci lines, where it turned into 61.8%.