Also known as the Golden Ratio, the Fibonacci sequence was introduced by Leonardo Pisano Bogollo (1170-1250), an Italian mathematician from Pisa, and can be listed as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610……

The sequence can be infinitely extended and contains many unique mathematical properties:

1. After 0 and 1, each number is the sum of the two prior numbers (1+2=3, 2+3=5, 5+8=13 8+13=21 etc…).

2. A number divided by the previous number approximates 1.618 (21/13=1.6153, 34/21=1.6190, 55/34=1.6176, 89/55=1.6181). The approximation nears 1.6180 as the numbers increase.

3. A number divided by the next highest number approximates .6180 (13/21=.6190, 21/34=.6176, 34/55=.6181, 55/89=.6179 etc….). The approximation nears .6180 as the numbers increase. This is the basis for the 61.8% retracement.

4. A number divided by another two places higher approximates .3820 (13/34=.382, 21/55=.3818, 34/89=.3820, 55/=144=3819 etc….). The approximation nears .3820 as the numbers increase. This is the basis for the 38.2% retracement. Also, note that 1 – .618 = .382

5. A number divided by another three places higher approximates .2360 (13/55=.2363, 21/89=.2359, 34/144=.2361, 55/233=.2361 etc….). The approximation nears .2360 as the numbers increase. This is the basis for the 23.6% retracement.

Among of the key numbers mentioned, 1.618 refers to the Golden Ratio or Golden Mean. The inverse of 1.618 is .618. These ratios can be found throughout nature, architecture, art, and biology.

**Trading with Fibonacci Retracement Levels**

In order to find Fibonacci retracement levels, we have to identify the recent significant Swing High and Swing Low, connect them and those levels are automatically presented.

Fibonacci retracement levels are providing traders or investors with an alarm of a potential trend reversal, in either resistance areas for uptrend or support areas for downtrend. That is to say, retracements are based on the prior move. A bounce is expected to rally a portion of the prior decline, while a correction is expected to pull back a portion of the prior advance.

With retracement levels applied individually, once a pullback starts, chartists can keep a close eye on those levels. As the correction approaches these retracements, investors should be more alert for a potential bullish reversal, as shown in Figure 1.

On the other hand, once a bounce begins, chartists need specific Fibonacci retracement levels for monitoring. As the correction approaches these retracements, investors should become more alert for a potential bearish reversal, as shown in Figure 2.

Based on depth, a 23.6% retracement can be considered to be relatively shallow. Such retracements would be appropriate for flags or short pullbacks. Retracements in the 38.2%-50% range would be considered moderate. Even though deeper, the 61.8% retracement can be referred to as the golden retracement.

We should keep in mind that retracement levels are not hard reversal points, rather they serve as alert zones for a potential reversal. In this case, it is recommended that traders should employ other technical indicators to identify or confirm a reversal.