Fibonacci Numbers

Fibonacci Numbers

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Both the Fibonacci Number Series and the Golden Ratio occur frequently in nature. Although these ideas were developed independently and in different millennia, they are related by the fact that, as the series grows, the ratio between successive numbers gets closer and closer to the Golden Ratio. Both phenomena (especially the Golden Ratio) can also be seen frequently in the movement of security prices, which is why we are interested in them.

This section provide a brief account of where these numbers come from. Feel free to skip it if you don't care for mathematics. However, if you want to know a little bit about these ubiquitous numbers, give it a try.

Leonardo of Pisa, who is more commonly known as Fibonacci (filius Bonacci, son of Bonacci), lived from 1170 until 1250. Born in Pisa (now in Italy), he was raised and educated in what is now Algeria. Considered one of the greatest mathematicians of the Middle Ages, he introduced Arabic numerals and the Hindu-Arabic decimal positional system into Europe.

Fibonacci is best known, however, for the series of numbers that arose from the following problem:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

This problem yields the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ..., which we call the Fibonacci Series. Note that each term in the series is the sum of the two terms that precede it.

The Fibonacci Series is interesting by itself. It has been studied extensively and has many manifestations, some of which can be found in the study of price movements.

What technical analysts have found more interesting, however, is the ratio between successive terms of the series. As the terms get larger, the ratio between each new term and its predecessor gets closer and closer to 1.6180339887..., which is usually shortened to 1.618. By the time the series reaches 55, the ratio is 1.618. There's no evidence that Fibonacci himself was aware of this, however; the earliest known reference was written by Pacioli in 1509.

This number is the Golden Ratio. It was used in the Great Pyramid at Gizeh, in Minoan architecture and in the Parthenon. It was extensively discussed by Euclid in his Elements, where he described it in terms of a line AB

that is divided at C in such a way that CB/AC = AC/AB (the line from your elbow (A) through your wrist (C) to your fingertips (B) is an example of such a line). Another way to express this relationship is to say that "the smaller line is to the larger what the larger is to the whole".

If we represent the length of CB by 1 and the length of AC by x, we can write the above relationship as:
   1/x = x/(x+1) or
   x + 1 = x² or
   x² - x - 1 = 0,
for which the solutions are (1 +

5)/2 and (1 -

5)/2, or 1.618... and -0.618... To explore these ideas further look here.

If we write 1.618... as Ø and we write term n of the Fibonacci Series as f_n, then the following series are equivalent (for sufficiently large values of n):

...	Ø^-4	Ø^-3	Ø^-2	Ø^-1	Ø⁰	Ø¹	Ø²	Ø³	Ø⁴	...
. . .	f_n-4 ------- f_n	f_n-3 ------- f_n	f_n-2 ------- f_n	f_n-1 ------- f_n	f_n ------- f_n	f_n ------- f_n-1	f_n ------- f_n-2	f_n ------- f_n-3	f_n ------- f_n-4	. . .
...	.146	.236	.382	.618	1	1.618	2.618	4.236	6.854	...

What the second row of the table shows is that when you take ratios of non-adjacent terms you still get Fibonacci ratios (ie, f_n/f_n-p = Ø^p). Note that for this series of ratios it is also true that each term is the sum of its two immediate predecessors..

Certain other numbers are also used:
   Ø⁰/2 = .5
   3Ø⁰/2 = 1.5
   2Ø⁰ = 2

Ø = 1.272
1/

Ø = .786