 |
 |
 |
 |
Fibonacci Numbers and the Pascal Triangle |
 |
Golden`s Section Formulas
|
|||||||||||||||||||||||||||||||||||||||||||||
|
Leonardo Fibonacci was born in Pisa, Italy, around 1175. His father was Guilielmo Bonacci, a secretary of the Republic of Pisa.His father was also a customs officer for the North African city of Bugia. Some time after 1192. Bonacci brought his son with him to Bugia.Guilielmo wanted for Leonardo to become a merchant and so arranged for his instruction in calculational techniques, escpecially those involving the Hindu - Arabic numerals which had not yet been introduced into Europe.
 Around 1200, Fibonacci returned to Pisa. Leonardo Fibonacci was the gratest European mathematician of the Middle Ages. He was the first to introduce the Hindu - Arabic number system into Europe. Leonardo wrote a book on how to do arithmetic in the decimal system, called "Liber abaci", completed in 1202. It describes the rules we are all now learn at elementary school for adding numbers, subtracting, multiplying and dividing.A problem in the third section of Liber abaci led to the introduction of the Fibonacci numbers :
A certain man put a pair of rabbits in a place surrounded on all sides by walls. 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?
By charting the populations of rabbits Fibonacci discovered a
number series from which one can derive the Golden Section. Here`s the beginning of the sequence :
French mathematician Edouard Lucas (1842 - 1891) gave the name Fibonacci numbers to this series and found many other important applications of them.
The Fibonacci numbers appear as leaf arrangements because the Fibonacci numbers form the best whole number approximations to the Golden Section.Deviding each number in the Fibonacci series by the one which preceeds, we will find the following series of numbers :
It also produces a ratio which stabilizes around the value of Phi :
There are interesting series of equations for Fibonacci numbers :
  In general case we have :
We have following relations between Fibonacci numbers and Golden Section :
  
Or in general case :
 There is a surprising connections with Golden Section and Fibonacci numbers. To ilustrate this we shall now introduce operator of the finite differences that associates the function
 with the function
 It is easy to verify that
 where D is operator of differentation.
We use the operator`s equation in formula for the Fibonacci numbers
 which is analogue with the well known equation:
 We have the identity :
 As will be seen, we usually deal with next operator`s equations for the Fibonacci and Lucas numbers :
 and  By using this equations, we can write :
 Now, we have the next series of equations:
 or in a general case :
 By using the identity for phi and Phi we can write:
| |||||||||||||||||||||||||||||||||||||||||||||
|
© 2001-2003 Radoslav Jovanovic translated: D.Filipovic created: January 2003. | |||||||||||||||||||||||||||||||||||||||||||||