Fibonacci

1. Fibonacci Leonardo Pisano

2. The start of a genius • We note that in a time interval of one thousand years, i.e. from 400 until 1400, then existed only one distinguished European mathematician, namely Leonardo of Pisa. Known as Fibonacci (that means son of Fibonacci), although he liked people called him “ Bigollo” that means “good for nothing”, he was born probably between 1170 and 1180 and died after 1240. • The father of Fibonacci, as secretary to the republic of Pisa, was sent to Bougie, Algeria, where Fibonacci received an excellent mathematic education. • In 1192 he was initiated into the theory of practice of business and particular calculating methods, including Indian calculating methods, based on the decimal system. Few years ago he extended his knowledge through travels in Egypt, Syria , Byzantium, Sicily and Provence.

3. His publications • In 1202 he published the book “Liber Abaci” where he presented the Indian numbers system, introducing the famous Fibonacci sequence: 1,1,2,3,5,8,13…. In the prefacy of this book the commented that his father was who thought him Arithmetic and gave support to study mathematics. In this book we can find algebraic methods, and rules for commercial practice. • In 1220 he wrote “Practica geometriae”. In 1225 “Flos”, in 1227“Liber quadratorum”. A lot of books were lost because in this time there were not printers, and books were made by hand. About “Liber quadratorum” we can find the first proof of the identity: • In other way, Fibonacci at some point in his work understood the importance of the negative number which he interpreted at “losses”.

4. Curiosities • In 1225 the emperor Frederic II postponed his departure for a crusade in order to find the time to organize a mathematical conference. Fibonacci attempted the conference and solved successfully all the suggested problems. Which two of these problems were: 1.- Find a rational number a/b such that the expression are squares of rational numbers. (solved using Diophantus). 2.- Solve the equation

5. How did the Fibonacci Sequence born? • Fibonacci numbers came up in relation to the following problem (Liber Abaci): Assume that a rabbit’s pregnancy lasts one month and that every female rabbit becomes pregnant at the beginning of every month starting from the moment that is one month old. Assume also that female rabbits always give birth to two rabbits, one male and one female. How many pairs of rabbits will exist on January 2, 1203if we start with a newborn pair on January 1, 1202. The number of rabbits increase as follows 1,1,2,3,5,8,13,21,34,55,… • Note that these sequence is represented several times in the nature, another example is the seeds of the plant helianthus are ordered in such a way that they form two winds of arcs starting from the center. The number of seeds located on each of these arcs is 21 and 13, which are successive Fibonacci numbers.

6. Fibonacci Sequence • The formula that gives the nth terms Fn of the sequence of Fibonacci is: We can see another form to give this sequence(less known):

7. Properties Of the Sequence • - The sum of the first n terms is: • - The sum of the n odd terms is: • - The sum of the n even terms is: • - The sum of the squares of the first n terms is: • - If m divide n then am divide an. • - Two consecutive numbers of Fibonacci Sequence are primes. • - And the property most important is that the ratio of consecutive Fibonacci numbers converges to the golden ratio, j, as the limit: