**The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa,Italy (see Pisa on Google Earth), the city with the famous Leaning Tower, about 1175 AD.**

**Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the present-day Algerian town of Béjaïa, (see Bejaia on Google Earth ) formerly known as Bugia or Bougie, where wax candles were exported to France. They are still called "bougies" in French.**

**So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others.**

**D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250).**

**By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci! Vinci was just a few miles from Pisa on the way to Florence, but Leonardo da Vinci was born in Vinci in 1452, about 200 years after the death of Leonardo of Pisa (Fibonacci).**

Fibonacci memorials to see in Pisa |

## The Fibonacci Numbers

# In Fibonacci's *Liber Abaci* book, chapter 12, he introduces the following problem (here in Sigler's translation - see below):

How Many Pairs of Rabbits Are Created by One Pair in One Year

A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.

**He then goes on to solve and explain the solution:**

Because the above written pair in the first month bore, you will double it; there will be two pairs in one month.

One of these, namely the first, bears in the second montth, and thus there are in the second month 3 pairs;

of these in one month two are pregnant and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month;

...

there will be 144 pairs in this [the tenth] month;

to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.

To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year.You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.

*beginning*1

*first*2

*second*3

*third*5

*fourth*8

*fifth*13

*sixth*21

*seventh*34

*eighth*55

*ninth*89

*tenth*144

*eleventh*233

*end*377

The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.

** **e.g. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.

These numbers possess a number of interrelationships, such as the fact that any given number is approximately 1.618 times the preceding number.

**The first two numbers in the series are one and one. To obtain each number of the series, you simply add the two numbers that came before it. In other words, each number of the series is the sum of the two numbers preceding it.**

**Note: Historically, some mathematicians have considered zero to be a Fibonacci number, placing it before the first 1 in the series. It is known as the zeroth Fibonacci number, and has no real practical merit. We will not consider zero to be a Fibonacci number in our discussion of the series.**

Binet's Formula |

**The Fibonacci series is defined recursively. That is, in order to find each term of the series using the definition, you have to find all the terms that precede it. This makes finding the nth term very difficult for large values of n, as you must find every term that comes before.**

**However, there could be a way to find Fibonacci numbers**

*without*using the definition. If this were possible, one would be able to find the nth term of the series simply by plugging n into a mathematical formula.**In 1843, Jacques Philippe Marie Binet discovered just such a formula for finding the nth term of the Fibonacci series. The formula itself looks like this:**

**The proof of this equation is dependant on concepts we have not yet explored, and will be given in full in a later lesson.**

**Binet's formula involves two very special numbers, the two numerical expressions within the lowest sets of parentheses. Remember these two numbers: they will become very important in later sections.**

**Related Links**:

**Discover how this amazing ratio, revealed in countless proportions throughout nature, applies to the financial markets. Fibonacci And The Golden Ratio**

Advanced Fibonacci tools can provide accuracy without sacrificing readability. High-Tech Fibonacci

We break down this indicator into simple, easy-to-understand pieces so you can profit. Make Money With The Fibonnaci ABC Pattern

Uncover the history and logic behind this popular trading tool. Taking The Magic Out Of Fibonacci Numbers

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Advanced Fibonacci tools can provide accuracy without sacrificing readability. High-Tech Fibonacci

We break down this indicator into simple, easy-to-understand pieces so you can profit. Make Money With The Fibonnaci ABC Pattern

Uncover the history and logic behind this popular trading tool. Taking The Magic Out Of Fibonacci Numbers

Find out how volume, the Aroon indicator and Fibonacci numbers can improve your profits. 3 Technical Tools To Improve Your Trading

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