I make extensive use of Fibonacci numbers and ratio's in my trading endeavours.   I was mentored by Larry Pesavento (of Gartley & Butterfly patterns fame) and his colleague, Costas Vayonis.

Leonardo Da Pisa Fibonacci

Have you ever wondered where we got our decimal numbering system from? The Roman Empire left Europe with the Roman numeral system which we still see, amongst other places, in the copyright notices after TV programmes (1997 is MCMXCVII).

The Roman numerals were not displaced until the 13th Century AD when Fibonacci published his Liber abaci which means "The Book of Calculations".

Leonardo Fibonacci c1175-1250.

Fibonacci, or more correctly Leonardo da Pisa, was born in Pisa in 1175AD. He was the son of a Pisan merchant who also served as a customs officer in North Africa. He travelled widely in Barbary (Algeria) and was later sent on business trips to Egypt, Syria, Greece, Sicily and Provence.

In 1200 he returned to Pisa and used the knowledge he had gained on his travels to write Liber abaci in which he introduced the Latin-speaking world to the decimal number system. The first chapter of Part 1 begins:

These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated.

Fibonacci sequence

Fibonacci is perhaps best known for a simple series of numbers, introduced in Liber abaci and later named the Fibonacci numbers in his honour.

The series begins with 0 and 1. After that, use the simple rule:

Add the last two numbers to get the next.

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,...

You might ask where this came from? In Fibonacci's day, mathematical competitions and challenges were common. For example, in 1225 Fibonacci took part in a tournament at Pisa ordered by the emperor himself, Frederick II.

It was in just this type of competition that the following problem arose:

Beginning with a single pair of rabbits, if every month each productive pair bears a new pair, which becomes productive when they are 1 month old, how many rabbits will there be after n months?

(Answer: Imagine that there are xn pairs of rabbits after n months. The number of pairs in month n+1 will be xn (in this problem, rabbits never die) plus the number of new pairs born. But new pairs are only born to pairs at least 1 month old, so there will be xn-1 new pairs:   xn+1 = xn + xn-1
Which is simply the rule for generating the Fibonacci numbers.)

The Golden Section

A special value, closely related to the Fibonacci series, is called the golden section. This value is obtained by taking the ratio of successive terms in the Fibonacci series:

Ratio of successive Fibonacci terms.

If you plot a graph of these values you'll see that they seem to be tending to a limit. This limit is actually the positive root of a quadratic equation (see box) and is called thegolden section, golden ratio or sometimes the golden mean.

The golden section is normally denoted by the Greek letter phi. In fact, the Greek mathematicians of Plato's time (400BC) recognized it as a significant value and Greek architects used the ratio 1:phi as an integral part of their designs, the most famous of which is the Parthenon in Athens.

The Parthenon in Athens.