When traders use Fibonacci retracements and projections they are also, without always knowing it, using an element of Elliott Wave Theory. In the 1940′s, R. N. Elliott enhanced his initial Wave Theory tenets to include Fibonacci numbers, as he noticed that the mass psychology underlying the markets’ movements tended to repeat over time. Elliott related this repetition to an important sequence of numbers that the mathematician Fibonacci tied to the reproductive increase of a theoretical population of rabbits.

**Understanding the Fibonacci Sequence**

In a 1,200s book *Liber Abaci*, Leonardo Pisano Bigollo, an Italian who was also known as Fibonacci, derived the Fibonacci Sequence from a hypothetical question involving rabbit reproduction.

To generate the Fibonacci sequence, one can follow the following steps:

(1) Start with a pair of immature rabbits, one male and one female.

(2) Then you wait one month before breeding them.

(3) Then you wait one month for them to have a pair of babies – one girl and one boy.

(4) Then you wait one month when the original pair have babies, but their babies do not.

(5) Then you wait one month when the original pair have babies, and so do their first set of babies, but not their second set.

(6) Repeat the process over and over and over again until you get tired.

(7) Assume that no rabbits die.

(8) Count how many pairs of rabbits there are at the end of each month.

Steps (1) and (2) will generate the first two numbers in the Fibonacci sequence, which are 1 and 1.

Step (3) will generate the next number, which is 2 (one original pair and one pair of baby rabbits of the opposite sex).

Step (4) will generate the next number, which is 3 (one original pair, their first pair of babies, and a second pair of babies from the original pair).

Step (5) will generate the next number, which is 5 (one original pair, their first pair of babies, their second pair of babies, and a new pair of babies from the original pair and the first pair of babies.)

And so on…

Eventually you might note an important short cut, which is that you can simply compute the current number of rabbit pairs by adding together the previous two numbers. This generates the infinite Fibonacci sequence as follows:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…

**How the Fibonacci Sequence Relates to Elliott Wave Theory**

As this numerical sequence progresses, the ratio of one number divided by the next number in the sequence approximates 0.618, the so-called Golden Mean which inverted yields roughly 1.618. In addition, the ratio of one number to that seen two further in the sequence approaches 0.382, while three further approaches 0.236.

With the addition of the 0.5 ratio and the (1-0.236) = 0.764 ratio, the science of Fibonacci retracements was founded by Elliott when he observed that major market moves would be corrected by a degree that approximated a Fibonacci ratio of the move.

Elliott discovered that traders can compute a series of likely retracement targets, by multiplying the extent of the initial move by these ratios and then projecting them in the opposite direction from that move off of the end point price of the move.

**Fibonacci Projections**

Elliott also observed that impulsive waves in an unfolding trend tend to relate to each other either on a 1:1 or 1:2 ratio, or by adding the Fibonacci ratios, to get 1:1.236, 1:1.382, 1:1.5, 1:1.764, and so on.

This means that Fibonacci projections can be computed off of the end of the intervening correction to determine the likely extent of an impulse once another one or two impulses are already known.

This important technical analysis technique can provide forex traders with useful price objectives for both impulsive and corrective waves as they unfold over time.

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