Leonardo Fibonacci explained the exponential growth in nature through a well-known number sequence. In this sequence each number is the sum of the previous two consecutive numbers.
The sequence starts with 0 and 1 and goes on with: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, etc.
bonacci proved that this sequence can be manifested in the evolution of a natural growth phenomenon, as a solution to a mathematical problem based on the reproduction process of a pair of rabbits. However, the utility of the sequence lies in its fundamental properties, discovered in the eighteenth century:
1. When dividing the consecutive numbers of the sequence, ie: 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, etc., the result tends to approach the number 0,618.
2. When dividing the non-consecutive numbers of the sequence, ie, 1/2, 1/3, 2/5 3/8, 5/13, 8/21, etc., the result is the number 0,382.
3. The ratio of any number of the sequence in relation to the next lowest number, ie: 21/13, 13/8, 8/5, tends to be 1,618.
4. The ratio of any number of the sequence in relation to the next lowest non-consecutive number, ie: 21/8, 13/5, 8/3, tends to be 2,618.
The difference between the ratios and the result of the equation is greater when the the numbers used in the series are smaller. For example:
144/233 = 0,618: the result of the equation is a precise Fibonacci ratio.
144/89= 1.6179: with a smaller number the result of the equation only comes close to the Fibonacci ratio.
The 1,618 ratio and its inverse, the 0,618, were called by the ancient Greeks the 'golden ratio'.
Traders are not so interested in the numbers of the sequence as in the ratios between the numbers. These ratios can be used to identify support or resistance levels, find the targets for price movements, or even to determine the time period that a movement will last.
The most popular levels or ratios are:23.6%, 38.2%, 61.8% and 161,8%.
The 100% which is the full size of the movement to be analyzed and its half, the 50%, are commonly added to the Fibonacci levels, although they are not ratios of the sequence.
It's common to see a price correction towards a Fibonacci level after a clear trend has been developing. This doesn't mean that a price correction is to halt accurately at these levels, but most of the time price will slow down or interrupt the directional move and find temporary support and resistance at a Fibonacci level.
In this daily USD/CHF chart, the Fibonacci tool is anchored in the extremes of a trend that lasted about a month in late 2006. For nearly a year, prices were consolidated at the 23.6%, 38.2%; 61.8% levels. The 50% level has been omitted here for more clarity, but we 
encourage you to include it in your toolbox.
in the next chart, the anchorages of the Fibonacci tool were moved when the price reached new highs and lows. Observe how the old consolidation levels are mostly coinciding with the new levels, despite the fact 
they are dislocating the tool.
In the next chart, the upper anchorage is simply displaced in time but remains at the same price level. In contrast, the lower anchor has been placed at the new low reached by the price. Again, observe how the price takes into account the new Fibonacci levels.
By moving the tool to the newest extremes, we are also segmenting the chart into smaller sections, which in turn can be used as entry or exit points, or just as levels to adjust a trade size.
