**Leonardo Fibonacci**, also known as Leonardo of Pisa, was a 12th-century Italian mathematician. He is regarded as one of the greatest mathematicians of the Middle Ages and is famous for introducing the sequence of numbers that bears his name to Europe: the Fibonacci sequence.

Leonardo Fibonacci was born in Pisa around 1170 and travelled extensively during his life, visiting countries such as Egypt, Syria, Greece and Algeria. During his travels, Fibonacci studied mathematics and trade, learning from Arab merchants the Indo-Arabic number system that we still use today. Fibonacci is most famous for his work ‘Liber Abaci’ (The Book of the Abacus), published in 1202, which was one of the first texts to introduce Indo-Arabic numbers to Europe. In his book, Fibonacci explained how to use the Indo-Arabic numbering system and gave examples of arithmetic and business problems that could be solved with it.

## The Fibonacci sequence: its importance and infinite properties

The Fibonacci sequence is a series of numbers in which each successive number is the sum of the previous two numbers. The sequence starts with 0 and 1, and continues with 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and so on.

The Fibonacci sequence was introduced by Leonardo Fibonacci in his book ‘Liber Abaci’ in 1202, but the sequence had previously been described in India. Fibonacci used the Fibonacci sequence to describe the growth of an ideal population of rabbits, but the sequence has many other applications in mathematics, science and art.

The Fibonacci sequence is remarkable for many interesting properties, including its relationship to the golden ratio. The golden ratio, often referred to by the Greek letter φ (phi), is the ratio of two consecutive numbers in the Fibonacci sequence. The approximate value of φ is 1.61803398875… and appears in many forms in nature and art. For example, the ratio of the length of two segments of a snail shell or a tree branch often approximates φ.

Furthermore, the Fibonacci sequence has many other interesting mathematical properties and is found in various contexts, such as in number theory, geometry and cryptography.

## The golden ratio: what it is and where we find it

The golden ratio, also known as the golden section or the golden number, is the ratio of two quantities in which the ratio of the sum of the two quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller quantity. The approximate value of the golden ratio is 1.61803398875….

The golden ratio has been used in many works of art and architecture because it was considered an aesthetically pleasing ratio. Fibonacci discovered the golden ratio by studying the growth of an ideal population of rabbits, which followed a sequence similar to the Fibonacci sequence. Fibonacci noticed that the ratio between the number of rabbits in one generation and the number of rabbits in the next generation approached the golden ratio.

The golden ratio is also found in nature, for example in the structure of shells, the arrangement of sunflower seeds and the shape of galaxies. The golden ratio is also found in many works of art, such as the Parthenon in Athens and Leonardo da Vinci’s Mona Lisa.

The golden ratio has been the subject of extensive mathematical studies, as it has many interesting properties, such as being an irrational number and an algebraic equation of the second degree. It is also related to the Fibonacci succession, where the ratio between two consecutive numbers in the succession approaches the golden ratio.

## Fibonacci tracing: what it is and its uses in trend analysis

Fibonacci retracement, or Fibonacci retracement, is a technical analysis technique used in trading to determine the support and resistance levels of an asset. This technique is based on the Fibonacci sequence and the gold ratio.

Fibonacci retracement levels are the key levels where the asset is expected to rally after a significant upward or downward movement. The most common levels are 23.6%, 38.2%, 50%, 61.8% and 78.6% of the previous move. These levels represent the points at which traders might decide to enter or exit the market.

The Fibonacci retracement is used by traders to identify possible trend reversal points and to determine support and resistance levels. Traders use technical analysis tools to draw the trend line and then apply Fibonacci retracement levels to identify possible trend reversal points. If the retracement levels coincide with other technical analysis factors, such as previous price levels or trend lines, the probability of a trend change is higher.

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