FIBONACCI by ewghwehws


									Biography (1170-1250)

 Fibonacci is a short for the Latin "filius Bonacci" which means
 "the son of Bonacci" but his full name was Leonardo of Pisa, or
  Leonardo Pisano in Italian since he was born in Pisa (Italy).
  He was educated in North Africa where his father worked as a
       Fibonacci travelled widely with his father around the
                        Mediterranean coast .
In 1200 he returned to Pisa and used the knowledge he had gained
                  on his travels to write his books.

 Liber Abaci (1202), The Book of Calculation
 Practica Geometriae (1220), The Practice of
 Flos (1225), The Flower
 Liber Quadratorum (1225),The Book of Square
                     The Flower

the approximate solution of the following cubic equation:


in sexagesimal notation is , equivalent to
 The Book of Square Numbers

           Method to find Pythogorean triples:
When you wish to find two square numbers whose addition
    produces a square number, you take any odd square
number as one of the two square numbers and you find the
other square number by the addition of all the odd numbers
from unity up to but excluding the odd square number. For
 example, you take 9 as one of the two squares mentioned;
the remaining square will be obtained by the addition of all
 the odd numbers below 9, namely 1, 3, 5, 7, whose sum is
  16, a square number, which when added to 9 gives 25, a
                      square number.
                Liber Abaci

The book introduced the Hindu-Arabic number system into
 Europe , the system we use today, based on ten digits with
          its decimal point and a symbol for zero:


The book describes (in Latin) the rules for adding numbers,
           subtracting, multiplying and dividing.
Suppose a newly-born pair of rabbits ( male + female) are put in a
  field. Rabbits are able to mate at the age of one month so that at
  the end of its second month a female can produce another pair
  of rabbits. Suppose that our rabbits never die and that the female
  always produces one new pair ( male + female) every month
  from the second month on.

How many pairs will
there be in one year?!

At the end of the first month, they mate, but there is still only 1 pair
At the end of the second month the female produces a new pair, so
now there are 2 pairs of rabbits in the field.
At the end of the third month, the original female produces a second
pair, making 3 pairs in all in the field.
At the end of the fourth month, the original female has produced yet
another new pair, the female born two months ago produces her first
pair also, making 5 pairs…..
         We get the following sequence of numbers:

              1, 1, 2, 3, 5, 8, 13, 21, 34 ...

This sequence, in which each number is a sum of two
  previous is called Fibonacci sequence so there is the
  simple rule: add the last two to get the next!

The Fibonacci numbers are the sequence of numbers defined
  by the linear recurrence equation

             Fibonacci Rectangles

We start with two small squares of size 1 next to each other. On top
         of both of these we draw a square of size 2 (=1+1).
    We can now draw a new square - touching both a unit square
  and the latest square of side 2 - so having sides 3 units long; and
  then another touching both the 2-square and the 3-square (which
  has sides of 5 units). We can continue adding squares around the
  picture, each new square having a side which is as long as the
     sum of the latest two square's sides. This set of rectangles
  whose sides are two successive Fibonacci numbers in length and
   which are composed of squares with sides which are Fibonacci
             numbers, we call the Fibonacci Rectangles.
                  Fibonacci spirals

A spiral drawn in the squares, a quarter of a circle in each square.
One of the most fascinating things about the Fibonacci
  numbers is their connection to nature.

   the number of petals, leaves and branches
   spiral patterns in shells
   spirals of the sunflower head
   pineapple scales


The greatest European mathematician of the
  middle age, most famous for the Fibonacci
sequence, in which each number is the sum of
      the previous two and for his role
        in the introduction to Europe
                 of the modern
            Arabic decimal system.


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