Leonardo Fibonacci was an Italian mathematician born in the 12th century. While tackling a problem involving the prediction of rabbit population growth, he discovered the "Fibonacci numbers," which are a sequence of numbers where each successive number is the sum of the two previous numbers.
e.g. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.
As far as trading is concerned, this sequence is not all that important; rather, it is the quotient of the adjacent terms (e.g 144/89) that possesses an amazing proportion: roughly 1.618, or its inverse 0.618. This proportion is known by many names: the golden ratio, the golden mean, phi and the divine proportion, among others. So, why is this number so important? Well, almost everything in nature has dimensional properties that adhere to the ratio of 1.618, so it seems to have a fundamental function or presence. Therefore it would not be unimaginable for Fibonacci ratios to have some bearing on the market.
Interpretation of the Fibonacci numbers in technical analysis anticipates changes in trends as prices approach lines created by the Fibonacci studies. When used in technical analysis, the golden ratio is typically translated into three percentages: â€“ 38.2%, 50% and 61.8%. However, more multiples can be used when needed, such as 23.6%, 161.8%, 423% and so on.
There are four primary methods for applying the Fibonacci sequence to finance:
 Fibonacci retracements
Fibonacci retracements use horizontal lines to indicate areas of support or resistance. They are calculated by first locating the high and low of the chart. Then five lines are drawn: the first at 100% (the high on the chart), the second at 61.8%, the third at 50%, the fourth at 38.2%, and the last one at 0% (the low on the chart). After a significant price movement up or down, the new support and resistance levels are often at or near these lines. Levels above 100%, such as 161.8% can also be used to predict price extensions following the break of a significant high or low.
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 Fibonacci arcs
Finding the high and low of a chart is the first step to composing Fibonacci arcs. Then, with a compass-like movement, three curved lines are drawn at 38.2%, 50% and 61.8%, from the desired point. These lines anticipate the support and resistance levels, and areas of ranging.
 Fibonacci fans
Fibonacci fans are composed of diagonal lines. After the high and low of the chart is located, an invisible vertical line is drawn though the rightmost point. This invisible line is then divided into 38.2%, 50% and 61.8%, and lines are drawn from the leftmost point through each of these points. These lines indicate areas of support and resistance.
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 Fibonacci time zones
Unlike the other Fibonacci methods, time zones are a series of vertical lines. They are composed by dividing a chart into segments with vertical lines spaced apart in increments that conform to the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, etc.). These lines indicate areas in which major price movement can be expected.
 Fibonacci Sequence in scaling viewpoints
When designing a strategy or an indicator that has a look-back or look-forward feature and/or synthetically built scaled views of an algorithm(s). The Fibonacci sequence can be a powerful factor to insure a balanced vision of the bigger picture. As unlike dividing a value into equal parts (i.e. 2,4,6,8) the Fibonacci series produces a ray or a wedge that enables a porportional balanced and ever expanding scale of view. --Albruno 15:35, 10 June 2007 (BST)
Fibonacci levels are objective and consistent trading systems can be made based upon them. For instance one trader might always buy following a 61.8% retracement with a tight stop loss and an initial target of the 50% level above.
Simply because Fibonacci ratios appear often in nature, it does not follow that they will be visible in the market or have any influence on traders' decisions. There is a whiff of the self-fulfilling prophecy about them: they appear to "work" simply because a large enough number of traders are watching Fibonacci levels and act on them. As an increasing number of traders watch Fibonacci levels, any edge that might be gained by doing so is diminished.
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