Leonardo Pisano, better known by hs nickname Fibonacci, was an Italian mathematician born in around 1170 in the Republic of Pisa. He was educated in North Africa, where his father held a diplomatic post, and at some point was sent to study calculation with an Arabic master. He later went to Egypt, Syria, Greece and Sicily, where he studied different numerical systems and methods of calculation.

Fibonacci is widely considered to be the most talented Western mathematician of the Middle Ages. When his Book of Calculation first appeared in 1202, Hindu - Arabic numerals were known only to a few European intellectuals, through translations of 9th century Arabic writings. Through his book, Fibonacci was solely responsible for popularising the Hindu - Arabic numeral system to the Western world, and this was the same book in which Europe was introduced to his famous number sequence.

Part 1 of the book dealt with the notation. This explained the principle of price value, by which the position of a figure determines whether it is a unit of 10, 100 and so on, and demonstrating the use of these numerals in arithmetical operations. The techniques were then applied to practical problems such as profit margin, barter, money changing, the conversion of weights and measures, partnerships and interest. In time, Fibonacci was being presented with a series of problems to solve. A problem in the third section of 'Liber Abaci' - the Book of Calculation, led to the introduction of the Fibonacci numbers and the sequence for which he is best remembered. The problem was worded in this way ...

*''A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year, if it is supposed that every month each pair begets a new pair, which from the second month on becomes productive?'' *The resulting answer and sequence became the now famous Fibonacci sequence of numbers.

The sequence is 1,1,2,3,5,8,13,21,34,55, etc ... where each number in the sequence is obviously the sum of the 2 preceding numbers. Fibonacci himself omitted the first value of zero in Liber Abaci. However the numbers themselves proved to be not as important as the mathematical relationships, expressed as ratios, between the numbers in the series. One of the truly remarkable characteristics of the sequence is that each number is approximately 1.618 times greater than the previous number. This and other common relationships between every number in the series is the foundation of the common ratios used in retracement studies and technical analysis.

There are three key ratios to look out for. These are 23.6%, 38.2% and 61.8%. Although not officially a Fibonacci number, traders often look to the 50% level too. Fibonacci retracements are the most widely used of all the Fibonacci trading tools. This is partially due to their relative simplicity but also due to their applicability to almost any trading instrument. They can be used to identify and confirm support and resistance levels, place stop loss orders or target prices, and even act as a primary mechanism in a countertrend trading strategy. For example, they can be used to determine critical points that cause an asset's price to reverse. The direction of the prior trend is likely to then continue, once the price of the asset has retraced to one of the key ratios.

In technical analysis, a Fibonacci retracement is created by taking two extreme points, usually a major peak and trough on a stock chart, and dividing the vertical distance by a key Fibonacci ratio. Once these levels have been identified, horizontal lines are then drawn and used to identify possible support and resistance levels.

*The key Fibonacci ratio is 61.8%, also referred to as the ''Golden Ratio''. It is found by dividing any number in the series by the number that follows it. For example, 21 divided by 34 is 0.6176, and 55 divided by 89 equals 0.6179. *

*The 38.2% ratio is found by dividing any number in the sequence by the number that is two places to its right. For example, 55 divided by 144 equals 0.3819. *

*The 23.6% ratio is found by dividing any number in the sequence by the number that is three places to its right. For example, 8 divided by 34 equals 0.2352.*

For reasons that are not entirely clear, these ratios seem to play an important role in the stock market. However, the use of Fibonacci retracements is subjective. Different traders will use this technical indicator in different ways. The traders who are profitable using the retracements will verify its effectiveness, while those who lose money will say it is unreliable. Some will argue that technical analysis is a case of a self fulfilling prophecy. If traders are all using and watching the same levels or technical indicators, then the price action may reflect that fact. The use of Fibonacci studies is also somewhat subjective in that the trader must use highs and lows of their choice. When building trading systems, traders will choose to select a high and a low during a set period of time.

The underlying principle of any Fibonacci tool is a numeric anomaly that is not grounded in any logical proof. The ratios, integers, sequences and formulas derived from the Fibonacci sequence are only the product of a mathematically based irregularity. This is not inherently wrong, but it can be uncomfortable for traders who want to fully understand the rationale behind a trading strategy. Due to this, many if not most traders will use Fibonacci numbers and levels in conjunction with other forms of technical analysis. For example, traders may look for confirmation of a breakout from a Fibonacci retracement by looking at the OBV or RSI (the On Balance Volume and the Relative Strength Index). These studies may also be used in conjunction with chart pattern analysis, such as ascending triangles or flags, to calculate any 'take profit' or 'stop loss' points along the way.

The famous Fibonacci sequence moves towards a certain constant, irrational ratio. In other words, it represents a number with an endless, unpredictable sequence of decimal numbers, which cannot be precisely expressed. When reduced to three decimal places, it is quoted as 1.618 and referred to as the 'golden section' or the 'golden average'. In algebra, it is commonly indicated by the Greek letter 'Phi' or 'Pi' **π**** **= 1.618. As the Fibonacci sequence moves on, each new number will divide the next one, always coming closer and closer to the unreachable Phi. Fluctuations of the ratio around the value 1.618 for a lesser or greater value can also be seen when using Elliott Wave Theory.

Fibonacci's sequence has captivated mathematicians, artists, designers and scientists for centuries. Its astounding functionality in nature suggests its importance as a fundamental characteristic of the Universe. The emerging patterns and ratios can be seen from the microscale to the macroscale, through to biological systems and inanimate objects. Some examples in nature include plants expressing the sequence in their growth points, shells' proportional increases, leaves' veins, the human body and face proportions, DNA molecules, the family tree of honeybees, the geometry of crystals, animal skeletons, the arrangement of pine cones, pistils in flowers, the shape of a hurricane and even the spirals of the Milky Way, to name just a few.

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