Leonardo Fibonacci was a mathematician most well known for discovering a number sequence which now bears his name whilst solving a problem about the reproduction rates of rabbits.

Leonardo Fibonacci was a mathematician in the 13th century who in 1202 published the first of four books, Liber Abaci. Liber Abaci introduced the Hindu-Arabic number system (numbers 1 to 9 including zero) to Europe. The Hindu-Arabic number system then replaced the roman numeral system which was used during this time.

The initial problem which lead to the discovery of the Fibonacci number sequence was how fast could rabbits reproduce in an ideal environment. Fibonacci made a couple of assumptions during his investigation, firstly the rabbits never die and secondly each pair of rabbits does not give birth to more than one pair of rabbits each month.

The problem begins with a single pair of rabbits, one male and one female.

1. After the first month the pair of rabbits only produce one child, no second pair can be made so only the single pair can reproduce.
2. After the second month the first female produces a child and two pairs of rabbits exist.
3. After the third month, the original female produces another pair making 3 pairs in total.
4. After the fourth month the original pair has produced another pair and the female born two months ago produces her first pair. This make 5 pairs in total.

The sequence continues with the number of rabbits pairs being 1, 1, 2, 3, 5, 8, 13, 21, 34 etc at the end of each month.

To reproduce the sequence mathematically, we can add each number in the sequence with the number occurring before it in the sequence. Continuing in this fashion is the Fibonacci sequence of numbers.

1
1 + 1 = 2
2 + 1 = 3
3 + 2 = 5
5 + 3 = 8
8 + 5 = 13
13 + 8 = 21
21 + 13 = 34 and so on.

The Fibonacci series of numbers carry some very unique characteristics. One of them is the association with Phi. Phi is often referred to as the golden ratio, golden mean, golden number or golden section.

As an example, if we take the ratio of each Fibonacci number against the previous number in the series, i.e. 1/1 = 1, 2/1 = 2, 3/2 = 1.5 etc we can see the results closing in on a value. If we graph this relationship we can better understand what is happening:

The numbers clearly settle toward a certain number. The number is 1.618034, this is the number for Phi.

Author:

Standard and Poor's paper on Leonardo Fibonacci, the person who founded Fibonacci numbers and the significance of these numbers in the present financial market. Very interesting reading even for the person who has not much interest in the market.

Another interesting occurrence of Fibonacci numbers in nature is the relationship to proportional squares. Taking an example, we draw a series of squares starting with the size of the first numbers in the Fibonacci sequence.

Continuing to draw squares with a size equal to the next Fibonacci number in the sequence we end up with a picture of squares all placed together to from a larger rectangle. The following steps illustrate this concept:

1. Start with two squares of size 1 and 1 (the first numbers in the Fibonacci sequence)

2. Add a square of size 2 on top of these two squares, then add a square of size three to the right side of this square. The shape which is produced by the group of squares is known as the Fibonacci rectangle.

3. The process continues with a square representing the size of the next Fibonacci number being added to the Fibonacci rectangle, the new square will reside on the next clockwise face of the rectangle. This process continues indefinitely. The following picture shows the Fibonacci rectangle produced up to the 13 number.

The Fibonacci Spiral in nature

When a circle is drawn in each square in the Fibonacci rectangle so that a quarter circle appears in each square it produces a spiral. This is shown in the picture above on the right.

This spiral formation occurs in many places in nature. You may have seen this shape in various sea shells.

Author:

Great article which provides an overview of the magic of Fibonacci numbers and how to adapt them to trade analysis.

Learn more about the fascinating topic of Fibonacci numbers. This website explores in-depth the various fascinating characteristics of Fibonacci numbers.

Retracements
Markets exhibit price movement all the time, prices move and prices move down. It has been evidenced that markets display trend and counter trend moves which have relationships with various Fibonacci numbers.

The most significant Fibonacci related numbers are 61.8% and 38.2%. Both of these numbers are related to Phi, the golden number.

61.8% = 1/Phi (1.618033988)
38.2% = 1/Phi Squared

Both W.D.Gann and R.N.Elliott used Fibonacci numbers in their technical analysis methods. The most important concept to realise with Fibonacci numbers is the importance they play in identifying possible ending points for market retracements.

Taking an example, Elliott wave theory states that the market moves in pronounced waves, each of these waves is characteristed by a series of five ways moving upward in direction and three ways moving in downward direction.

Fibonacci analysts look at certain retracements equaling the two percentages of 61.8% and 38.2% to signify the turning points for the correction. This is best illustrated with an example.

These retracement levels are used extensively by Fibonacci analysts in particular Elliott wave analysts. An additional retracement percentage that is commonly used is 50%.

Time Zones
Fibonacci time zones are purely indicators placed on a chart representing the Fibonacci sequence of numbers. On a daily chart the Fibonacci time zones will place identifiers at 2, 3, 5, 8, 13 etc days. The concept behind this approach is to identify tops and bottoms correlating to certain Fibonacci numbers. This approach is more simplistic than retracement analysis above however the simplicity of this approach does not diminish from its validity.

It should be noted that the application of Fibonacci numbers in financial market forecasting is a subjective task. No indication of tops or bottoms are provided with these methods and most successful users of Fibonacci techniques combine the Fibonacci sequence of numbers with other indicators or oscillators.

Author:

Good article which provides an introduction to Fibonacci, Fibonacci numbers and the Golden Ratio.