FibonacFibonaccici

LeonardoLeonardo Pisano Pisano

The The startstart of of aa genius genius• We noteWe note that in a time interval of one thousand that in a time interval of one thousand

years, i.e. from years, i.e. from 400400 until until 14001400, then existed only one , then existed only one distinguished European mathematician, namely distinguished European mathematician, namely Leonardo of Pisa. Known as Fibonacci (that means Leonardo of Pisa. Known as Fibonacci (that means son of Fibonacci), although he liked people called son of Fibonacci), although he liked people called him “ Bigollo” that means “good for nothing”, he him “ Bigollo” that means “good for nothing”, he was born probably between was born probably between 1170 1170 and and 11801180 and died and died after after 12401240. .

• The father of Fibonacci, as secretary to the republic The father of Fibonacci, as secretary to the republic of Pisa, was sent to Bougie, Algeria, where Fibonacci of Pisa, was sent to Bougie, Algeria, where Fibonacci received an excellent mathematic education.received an excellent mathematic education.

• In In 11921192 he was initiated into the theory of practice he was initiated into the theory of practice of business and particular calculating methods, of business and particular calculating methods, including Indian calculating methods, based on the including Indian calculating methods, based on the decimal system. Few years ago he extended his decimal system. Few years ago he extended his knowledge through travels in Egypt, Syria , knowledge through travels in Egypt, Syria , Byzantium, Sicily and Provence.Byzantium, Sicily and Provence.

His His publicationspublications

• In In 12021202 he published the book “Liber Abaci” where he presented he published the book “Liber Abaci” where he presented the Indian numbers system, introducing the famous Fibonacci the Indian numbers system, introducing the famous Fibonacci sequence: sequence: 11,,11,,22,,33,,55,,88,,1313…. In the prefacy of this book the …. In the prefacy of this book the commented that his father was who thought him Arithmetic and commented that his father was who thought him Arithmetic and gave support to study mathematics. In this book we can find gave support to study mathematics. In this book we can find algebraic methods, and rules for commercial practice.algebraic methods, and rules for commercial practice.

• In In 12201220 he wrote “Practica geometriae”. In he wrote “Practica geometriae”. In 12251225 “Flos”, in “Flos”, in 12271227 “Liber quadratorum”. A lot of books were lost because in this time “Liber quadratorum”. A lot of books were lost because in this time there were not printers, and books were made by hand. About there were not printers, and books were made by hand. About “Liber quadratorum” we can find the first proof of the identity:“Liber quadratorum” we can find the first proof of the identity:

• In other way, Fibonacci at some point in his work understood the In other way, Fibonacci at some point in his work understood the importance of the negative number which he interpreted at importance of the negative number which he interpreted at “losses”.“losses”.

bc)+(ad+ bd)-(ac=)d+)(cb+(a 222222

CuriositiesCuriosities

• In In 12251225 the emperor Frederic II postponed his departure for the emperor Frederic II postponed his departure for a crusade in order to find the time to organize a a crusade in order to find the time to organize a mathematical conference. Fibonacci attempted the mathematical conference. Fibonacci attempted the conference and solved successfully all the suggested conference and solved successfully all the suggested problems. Which two of these problems were:problems. Which two of these problems were:

11.- Find a rational number .- Find a rational number a/ba/b such that the expression such that the expression

are squares of rational numbers. (solved using Diophantus).are squares of rational numbers. (solved using Diophantus).

22.- Solve the equation.- Solve the equation

2010x+2x+x 23

52(a/b)

HowHow did did thethe Fibonacci Fibonacci SequenceSequence born?born?

• Fibonacci numbers came up in relation to the following problem Fibonacci numbers came up in relation to the following problem (Liber Abaci):(Liber Abaci):Assume that a rabbit’s pregnancy lasts one month and that every Assume that a rabbit’s pregnancy lasts one month and that every female rabbit becomes pregnant at the beginning of every month female rabbit becomes pregnant at the beginning of every month starting from the moment that is one month old. Assume also that 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 rabbits always give birth to two rabbits, one male and one female.female.How many pairs of rabbits will exist on January How many pairs of rabbits will exist on January 22, , 12031203 if we start if we start with a newborn pair on January with a newborn pair on January 11, , 12021202. The number of rabbits . The number of rabbits increase as follows increase as follows 11,,11,,22,,33,,55,,88,,1313,,2121,,3434,,5555,…,…

• Note that these sequence is represented several times in the Note that these sequence is represented several times in the nature, another example is the seeds of the plant helianthus are nature, another example is the seeds of the plant helianthus are ordered in such a way that they form two winds of arcs starting 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 from the center. The number of seeds located on each of these arcs is arcs is 2121 and and 1313, which are successive Fibonacci numbers., which are successive Fibonacci numbers.

Fibonacci Fibonacci SequenceSequence

• The formula that gives the nth terms The formula that gives the nth terms FFnn of the sequence of of the sequence of Fibonacci is:Fibonacci is:

We can see another form to give this sequence(less We can see another form to give this sequence(less known):known):

1,1,, 21123 FFWhereFFF nnn

nnnF )

2

5/11)(5/1()

2

5/11)(5/1(

PropertiesProperties Of Of thethe Sequence Sequence

• - The sum of the first n terms is:- The sum of the first n terms is:

• - The sum of the n odd terms is:- The sum of the n odd terms is:

• - The sum of the n even terms is:- The sum of the n even terms is:

• - The sum of the squares of the first n terms is:- The sum of the squares of the first n terms is:

• - If m divide n then - If m divide n then amam divide divide anan. . • - Two consecutive numbers of Fibonacci Sequence are primes.- Two consecutive numbers of Fibonacci Sequence are primes.• - And the property most important is that the ratio of consecutive - And the property most important is that the ratio of consecutive

Fibonacci numbers converges to the golden ratio, Fibonacci numbers converges to the golden ratio, , as the limit:, as the limit:

1221 nn ffff

)2

5/11(lim 1

n

n

n F

F

nn ffff 21231

112242 nn ffff

12222

21 nn ffff